McGregor, who came through the academy system at Rangers, could return to Ibrox to face Graeme Murty's side next season.
Another ex Rangers player could be bound for one of the Ibrox side's Premiership rivals.
Hearts have already signed Kyle Lafferty and Steven Naismith this season, more than five years on from their Rangers exits.
Though a product of Rangers' academy, McGregor was born in the Scottish capital and a return there could make sense in the final years of his career.
But what are the Ibrox faithful saying about Hearts' alleged interest in the Hull City goalkeeper? Here's some of their tweets...
Might aswell go for weir, davis, bougherra and salim kerkar and really get the band back together y'know— Niall (@NiallPowers91) 13 March 2018
Get him to rangers.— james craig (@jwjcraig) 13 March 2018
Should change their name to Edinburgh Rangers— Yvonne Doyle (@deloresdelago) 13 March 2018
Great move Bill if moneys right...still a brilliant keeper— Gary H (@GH7542) 13 March 2018
If Jelavic and Bougherra pitch up next it’s time to ask questions— Heart & Hand (@ibroxrocks) 13 March 2018
Stuart Munro’s digging his boots out!— Tom McClure (@airman614) 13 March 2018
McGregor to hearts can't believe it. If that happens utterly shocking— Ally (@AlasdairMore) 13 March 2018
Why? He’s like 36 with zero sell on value and we have 2 fine GK’s.— Kirsten (@kirsten93_) 13 March 2018
36-year-old McGregor has been a regular in Hull's side this season, and missed his first league game on Saturday through injury - which saw David Marshall, who began his career at Rangers' rivals Celtic, replace him between the sticks.
The Scotland international has spent the past five years at the KCOM Stadium, having initially moved to Turkish giants Besiktas after Rangers' 2012 demotion to the fourth tier.
What are your thoughts on McGregor's potential summer switch?