It was reported last week that Stoke would allow the 35-year-old to leave on a free transfer.
Manchester United want the keeper to cover their two current keepers, David de Gea and Sergio Romero.
It would be likely that Grant would become their third choice, with Joel Perriera being sent on loan for more experience.
Grant signed a deal with Stoke in January 2017 after a move from Derby.
He was initially brought in as cover for the then-injured Jack Butland.
He made 30 appearances for Stoke in his first season, but only featured three times in the Premier League last season.