Having left Swansea City in June, the Scotland internationalist has yet to decide where his future lies. He can join any club outwith a transfer window as an out-of-contract player.
Hearts would like to make him their eighth new signing since the end of last season but must wait to learn whether he will come back north or pursue a deal in England.
The 26-year-old is attracting a number of clubs in different leagues around Britain and is aware of interest from Tynecastle Park. Hearts would like another forward to add to their squad and have been monitoring McKay for some time.
If he returns to Scotland it is understood a move to Gorgie would be high on his list of preferences.
Hearts finished their transfer window business on Tuesday with the arrival of Australian midfielder Cameron Devlin from Newcastle Jets. He joined other summer recruits Taylor Moore, Ben Woodburn, Beni Baningime, Alex Cochrane, Josh Ginnelly and Ross Stewart.
The Edinburgh club attempted to bring in an extra forward who can play out wide before the transfer deadline, however they wanted the right candidate rather than someone who would only fill a squad place.
McKay fits many of their requirements as a free agent with experience at the top level. He left Rangers in 2017 to join Nottingham Forest for £500,000 before moving on to Swansea 12 months later.
He spent last season on loan at Fleetwood Town and was released by Swansea at the end of the campaign when his contract expired.