Yes, even though, to be honest, nearly every SID in Germany practically ends when passing though the MRVA, where one gets a direct to some enroute waypoint…
In the US, either ATC or the pilot may initiate a visual approach under the required conditions. Here is what controllers are instructed on the issue:
7-4-1. VISUAL APPROACH
A visual approach is an ATC authorization for an aircraft on an IFR flight plan to proceed visually to the airport of intended landing; it is not an instrument approach procedure. Also, there is no missed approach segment. An aircraft unable to complete a visual approach must be handled as any go-around and appropriate separation must be provided.
7-4-2. VECTORS FOR VISUAL APPROACH
A vector for a visual approach may be initiated if the reported ceiling at the airport of intended landing is at least 500 feet above the MVA/MIA and the visibility is 3 miles or greater. At airports without weather reporting service there must be reasonable assurance (e.g. area weather reports, PIREPs, etc.) that descent and flight to the airport can be made visually, and the pilot must be informed that weather information is not available.
PHRASEOLOGY -
(Ident) FLY HEADING OR TURN RIGHT/LEFT HEADING (degrees) VECTOR FOR VISUAL APPROACH TO (airport name).
(If appropriate)
WEATHER NOT AVAILABLE.
NOTE – At airports where weather information is not available, a pilot request for a visual approach indicates that descent and flight to the airport can be made visually and clear of clouds.
7-4-3. CLEARANCE FOR VISUAL APPROACH
ARTCCs and approach controls may clear aircraft for visual approaches using the following procedures:
NOTE – Towers may exercise this authority when authorized by a LOA with the facility that provides the IFR service, or by a facility directive at collocated facilities.
a. Controllers may initiate, or pilots may request, a visual approach even when an aircraft is being vectored for an instrument approach and the pilot subsequently reports:
1. The airport or the runway in sight at airports with operating control towers.
2. The airport in sight at airports without a control tower.
b. Resolve potential conflicts with all other aircraft, advise an overtaking aircraft of the distance to the preceding aircraft and speed difference, and ensure that weather conditions at the airport are VFR or that the pilot has been informed that weather is not available for the destination airport. Upon pilot request, advise the pilot of the frequency to receive weather information where AWOS/ASOS is available.
PHRASEOLOGY -
(Call sign) (control instructions as required) CLEARED VISUAL APPROACH RUNWAY (number);
or
(Call sign) (control instructions as required) CLEARED VISUAL APPROACH TO (airport name)
(and if appropriate)
WEATHER NOT AVAILABLE OR VERIFY THAT YOU HAVE THE (airport) WEATHER.
c. Clear an aircraft for a visual approach when:
1. The aircraft is number one in the approach sequence, or
2. The aircraft is to follow a preceding aircraft and the pilot reports the preceding aircraft in sight and is instructed to follow it, or
NOTE – The pilot need not report the airport/runway in sight.
3. The pilot reports the airport or runway in sight but not the preceding aircraft. Radar separation must be maintained until visual separation is provided.
d. All aircraft following a heavy jet/B757 must be informed of the airplane manufacturer and/or model.
EXAMPLE –
“Cessna Three Four Juliet, following a Boeing 757, 12 o’clock, six miles.”
or
“Cessna Three Four Juliet, following a Seven fifty seven, 12 o’clock, six miles.”
e. Inform the tower of the aircraft’s position prior to communications transfer at controlled airports. ARTS/STARS functions may be used provided a facility directive or LOA specifies control and communication transfer points.
f. In addition to the requirements of para 7-4-2, Vectors for Visual Approach, and subparas a, b, c, d, and e, ensure that the location of the destination airport is provided when the pilot is asked to report the destination airport in sight.
g. In those instances where airports are located in close proximity, also provide the location of the airport that may cause the confusion.
EXAMPLE -
“Cessna Five Six November, Cleveland Burke Lakefront Airport is at 12 o’clock, 5 miles. Cleveland Hopkins Airport is at 1 o’clock 12 miles. Report Cleveland Hopkins in sight.”
When would you use/be given a “contact approach”? This is wprobably what Europeans mean by a Visual approach….ie only the pilot can initiate….in the US ATC can give you a visual approach whereas as AlanB points out they cannot in Europe…it has to be requested…
I have never actually used a contact approach, but would only use it if I was intimately familiar with the airport and surroundings, probably just my home base. It permits the pilot to fly under visual conditions that may be less than the basic VFR conditions and that don’t meet the requirements for a visual approach. Also, the airport does not need to be in sight in order to fly the contact approach, the pilot must remain in visual conditions, which can be as low as 1 SM and clear of clouds.
I am the opposite of an expert on European IFR, but I was told “expect visual approach” just this week at EKRN. Maybe the controller just did that to make me feel more at home 
A visual approach would be my default expectation in the US, weather permitting.
The one time I’ve used a contact approach in the US was when there was a shallow deck of cloud with bases 300 ft above an airport where the lowest MDH is 600 ft. On the first approach to the airport (on an IAP, ending in a missed approach), it became clear that if we approached the airport from a different direction, we’d be able to slide in under the layer. It helped that the direction we would have to approach from was aligned with the runway (whereas the approach courses on the IAPs are at a large angle) and that the layer stopped pretty much right at the airport fence. Since we had to maneuver a ways from the missed approach holding point before we’d be able to see the airport, a visual approach wouldn’t have worked, but, as NCYankee said, that was fine for a contact approach.
Visual approaches are most certainly not expected to use VFR arrival procedures! As several people have pointed out, a visual approach is flown under IFR. (You can actually fly a visual approach with lower than VMC flight visibility — down to 800 m, but how you do that in practise, I’ve not be able to figure out…)
Visual arrival routes in control zones are usually designed to be procedurally separated from IFR traffic, so it would not be a good idea to try to follow them under IFR.
There are cases where specific visual approach procedures are published, but it is unusual.
I have accepted a visual departure out of Köln which was offered by ATC and avoided me having to fly the full SID.
When you get asked that in Germany, the real question is “do you want to depart or prefer to burn your fuel at the holding point?”. So always answer yes. It is also very common to take a creative interpretation of VMC during such manoeuvers (the dreaded MRVA).
I was actually asked after getting airborne but I know what you say. Having flown out of Egelsbach I understand that VMC in Germany is “flexible”.
Visual approaches are most certainly not expected to use VFR arrival procedures! As several people have pointed out, a visual approach is flown under IFR. (You can actually fly a visual approach with lower than VMC flight visibility — down to 800 m, but how you do that in practise, I’ve not be able to figure out…)
What exactly is a VFR arrival procedure in controlled airspace? There is no single “standard” procedure, you can do whatever you get a clearance to do, and it varies between airports and weather, traffic etc. The only place you have something that can be called a “standard” VFR arrival procedure, is at unattended airports (no radio, no nothing), places where IFR is out of the question. And it’s not even a “standard”, more like a customary thing when no other information is available. Besides, 800 m visibility sounds very helicopter to me, in VMC.
unattended airports (no radio, no nothing), places where IFR is out of the question.
Even ENHA once used to have an NDB approach available even outside of ATS hours 
… In France there are many IFR approaches available outside of ATS hours, possibly even at airfields w/o ATS. With RNAV (GNSS) approaches we may see more of that. In the US too there are numerous airfields, even with no ATS at all, with IFR approaches.
In France though, if no ATS you are supposed to comply with the VFR traffic pattern at the conclusion of your approach. Straight-in is not according to the book.
