McGregor is one of several Hull City stars whose contracts expire this summer.
The Edinburgh side have already tied up a summer deal for St Johnstone striker Steven MacLean, and are said to want up to 10 new players ahead of next season.
And it is claimed that McGregor - or Cambridge United striker Uche Ikpeazu - could be next through the door, despite Nigel Adkins' desperation to keep hold of the 36-year-old, whose contract expires in June.
The Tigers boss told the Hull Daily Mail on Friday: "Allan knows I want him to be here. I’ve told him that.
"As a character and a goalkeeper, I can’t speak highly enough of him.
"He’s been outstanding this season. I pick him every week. Greggs was third choice at the start of the season, so I’m led to believe, but since he was put in the team he’s not looked back."
Edinburgh native McGregor - who played under Hearts boss Craig Levein for Scotland - began his career at Glasgow Rangers.
After leaving Ibrox in 2012, the three-time Scottish Premiership champion spent a season at Besiktas before heading to the KCOM Stadium five years ago.
McGregor may not be the only player to depart Hull for nothing this summer, with a string of key players out of contract.
According to reports last week, one of those - striker Abel Hernandez - has been offered a three-year deal by Hearts' league rivals, Celtic.