The 30-year-old has been part of the coaching staff at Old Trafford since Ole Gunnar Solskjaer brought him in last year.
Ramsay is highly rated as a coach at Manchester United and the club want him to be part of Erik ten Hag’s staff as well.
The Dutchman is also keen to have him as part of his staff but interest from Barnsley is said to have complicated things.
The League One club have looked at an approach for the 30-year-old in their pursuit of a new manager.
But it has been claimed that Ramsay will stay at Manchester United as part of Ten Hag’s backroom team.
Manchester United are ready to hold talks with Ramsay if Barnsley make an official approach for him.
However, there is no confidence that the 30-year-old is now set to stay at Old Trafford.
Manchester United did allow Kieran McKenna to leave when Ipswich Town made an approach in December last year.