And Schneiderlin arrived at the club's Finch Farm training complex early on Wednesday afternoon as the move gathered pace.
The France international was brought to United by Louis van Gaal in a £24m deal in the summer of 2015 but has failed to win favour since Jose Mourinho took over last summer.
It is understood the fee could rise to £24m with add-ons, with Schneiderlin available to face Manchester City, live on Sky Sports on Super Sunday.
After starting 25 Premier League games and making four substitute appearances during his debut campaign at Old Trafford, Schneiderlin has been limited to three brief substitute appearances in the league this season.
"I am sad and happy, sad because I like him and he could be an option for us, happy because this is what he wants, to play every game and be important in the team."
Last month, Mourinho confirmed he would not stand in Schneiderlin's way if he wished to leave in search of first-team football and United received an offer that matched their valuation of the 27-year-old.
When asked about Schneiderlin, Mourinho had said: "Great professional. Fantastic boy, very honest, very open and he opened his heart a couple of times and my answer is simple: if he's playing regular with me I have the right to say no way, if he's not playing regular I have no right to tell a player that if he wants to be happy you're going nowhere.
"So my answer was if the offer is right and our board thinks the offer is a good offer in relation to a very good player like Morgan I would not stop him to go."
West Brom had also bid for Schneiderlin, while Everton have also been credited with a continued interest in United's Memphis Depay.