The Daily Record and Scottish Sun are both reporting that Rangers have made a two-year contract offer to their former player, who is set to become a free agent when his contract with Hull City expires this summer.
McGregor has been the Championship side’s No.1 this term, keeping international team-mate David Marshall in reserve, though a new contract was not forthcoming earlier this campaign.
It emerged recently that Hull were finally looking to tie him to a new deal, though McGregor looks set to reject that offer in favour of a return to Scotland.
Hearts have also been credited with an interest in the player.
McGregor spent 11 years at Ibrox earlier in his career. He rose through the youth ranks and would tally 278 appearances before leaving for Besiktas in 2012.