The 32-year-old is on United’s list of emergency targets following their problems in the left-back position.
Luke Shaw will be out for a few months, while Tyrell Malacia has been injured since the early part of pre-season.
United have drawn up an emergency list of left-backs to sign before the close of the transfer window on Friday.
However, Alonso is not keen on a return to the Premier League and wants to stay at the Catalan club.
Alonso joined Barcelona last summer from Chelsea and has mostly played back-up to Alejandro Balde at left-back and covered in central defence.
He has struggled for game time at his new club, starting only 11 matches last season in La Liga. He has started only once this season.
While the move to United might see him get more game time, he is not ready to change surroundings.
Alonso knows the Premier League very well from his six years at Chelsea. He made 212 appearances, bagging 29 goals and 23 assists for the Blues.
However, United will have to look elsewhere with the Spaniard refusing to move.
The Red Devils still have a few options to sift through. They might have even added another target to the list, with AS Roma full-back Leonardo Spinazzola a possible target.
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