Manager Eddie Howe is a fan of the central midfielder, who has played for England at under-20 level.
The Cherries are willing to pay £1 million to United to secure him on loan until the end of the season.
Pearson is currently on loan at League One side Barnsley, where he has played 26 times in all competitions.
The chance to play in the Premier League is not one he will want to miss out on, especially with opportunities likely to be limited by the central midfield competition at United.
Another tough blow for Barnsley
Barnsley are in 21st place in League One and will be disappointed at the prospect of losing Pearson for the season.
This week they saw Newcastle United recall striker Ivan Toney, and weeks ago Arsenal brought back midfielder Dan Crowley from his loan with the Tykes, after several impressive performances.