United boss Jose Mourinho is looking to provide cover for David de Gea and Sergio Romero in his squad at Old Trafford next season.
And the Telegraph say the man he has chosen in 35-year-old veteran Grant, who is available without a fee after the end of his contract at the Bet365 Stadium.
The idea will be for Grant to come in as the number three keeper, ensuring there is enough strength in depth behind De Gea and Romero in the first team squad.
United have highly-rated Portugal youngster Joel Pereira, 21, as third choice at present but want to send him out on loan for a season to gain experience.
And with their other keeper Sam Johnstone expected to leave this summer having impressed on loan at Aston Villa last season, it has been decided someone with experience is the best option to bring in as cover.
Grant has enjoyed a good career in the professional game, most of which has been spent outside the Premier League.
The majority of his 500-plus professional appearances have come at clubs like Derby, Burnley and Sheffield Wednesday.
But he finally got his chance to perform in the top division at Stoke, when he stepped in to replace the injured Jack Butland two seasons ago.
He made 28 Premier League appearances in the 2016-17 campaign, impressing with his ability and experience.
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And though Butland returned to full fitness last season to reclaim his spot, Grant's performances have been enough to convince United he is the ideal player to bring in.
It is expected he will sign a one-year contract, with an option to extend should United continue to need similar cover the following year.
United keeper Johnstone, 25, looks to be heading for the exit although Villa's hopes of making his move permanent have been hit by their financial problems.
But relegated Stoke and West Brom could both be in the market for a new number one over the summer, with Butland and Ben Foster attracting Premier League interest.
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