The 35-year-old will reportedly join Manchester United on a free transfer.
Sky Sports has reported that the 35-year-old would be joining United on a free transfer as cover for David de Gea and Sergio Romero.
After having initially joined Stoke City on loan as cover for Jack Butland, who was injured at the time, Grant signed on a permanent basis in January 2017 and appeared 30 times for the Potters that season.
The veteran goalkeeper put together some impressive displays for the Potters, including an outstanding performance in a 1-1 draw against United in October 2016, during Mourinho's first season at the club.
Last season Butland returned to full fitness and Grant appeared just three times for Stoke.
Grant's arrival would likely allow highly rated 22-year-old Joel Pereira to go out on loan and get a full season of first-team football in away from Old Trafford, with Dean Henderson already out on loan at Sheffield United and Sam Johnstone looking likely to join West Bromwich Albion permanently.
The news of the potential signing has not gone down well with a section of the United faithful, with many still unhappy with the club following the saga concerning Marouane Fellaini's new contract.
This our banter era, officially. Someone better check up on Jose, see if he's okay, worried he might have been the coke with good ol' Diego— Jordan Henshall (@JordyHensh07) June 27, 2018
What is going on .. first felliani and this !!?? We need to be signing top class talents not this crap— Nikhil Dhorepatil (@Nikhil689908) June 28, 2018
Jose’s lost it ffs— Dan (@DanielUtd) June 27, 2018
So we are signing someone to sit on their arse and get paid. So all our young keepers won't get a sniff of first team match— Reddevil76 (@Topps1765) June 28, 2018
Fellaini and Darmain staying at Utd and Lee Grant coming. I'm baffled.— James. (@FredSZN) July 2, 2018
Another no name to help us stay below city— bradfordroadcarsales (@AlecDod021) July 2, 2018
Richard Wright not knocking about anymore no?— Stephen™ ® (@Stephen_____B) June 27, 2018