The report claims that the 36-year-old is set to pen a two-year-contract with the Gers this week.
McGregor is entering the final few weeks of his contract at the KCOM stadium, and the Scottish international, who has 35 cap to his name, could be set to leave Hull City on a free transfer.
The veteran stopper is keen to return to Ibrox to end his career at his former club.