FWIW, the UK book (Manual of Air Traffic Services Part 1) says
12 Visual Approach
12.1 To expedite traffic at any time, IFR flights may be authorised to execute visual approaches if the pilot reports that he can maintain visual reference to the surface and
a) the reported cloud ceiling is not below the initial approach level; or
b) the pilot reports at any time after commencing the approach procedure that the visibility will permit a visual approach and landing, and a reasonable assurance exists that this can be accomplished.
12.2 Standard separation shall be effected between such aircraft and other IFR and/or SVFR aircraft.
This is a bit ambiguous because it doesn't say who should request the visual approach - personally, I will always wait for the pilot to ask but there are plenty of ways to let the pilot know that a request would be approved.