Manchester United have agreed a deal worth an initial £24m with Southampton for the France midfielder Morgan Schneiderlin as Louis van Gaal's significant reshaping of his squad gathers pace.

Schneiderlin travelled to Carrington on Sunday evening to undergo a medical, his tests undertaken immediately after those for Bastian Schweinsteiger whose own £14.4m move from Bayern Munich has now been finalised. United hope to have both players on board their flight bound for Seattle on Monday as they embark on a pre-season tour of the United States.

Van Gaal had identified Schneiderlin, who had also attracted tentative interest from Arsenal and Tottenham Hotspur, as a prospective long-term successor to Michael Carrick in United midfield's after his impressive displays for Southampton last season, when he made 28 Premier League appearances, two as a substitute, in a team who finished seventh. An initial bid of £20m from United was turned down last month, though Southampton's manager, Ronald Koeman, admitted on the squad's return for pre-season training that "every player has his price".

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It is understood that add-ons written into the deal could bring the total fee payable for Schneiderlin closer to Southampton's long established £27m valuation.

Meanwhile, Koeman suggested he was planning for life without the 25-year-old midfielder, who had risen through the divisions with Southampton since arriving from Strasbourg as a teenager. Koeman is expected to secure the Feyenoord midfielder, Jordy Clasie, as a replacement with the Dutch club having omitted the player from a pre-season friendly against Fortuna Düsseldorf on Sunday with talks over his future at an advanced stage.

Schneiderlin will link up in United's midfield with Schweinsteiger next season after the German international's arrival from Bayern. The World Cup winner flew in to Manchester on Sunday to undergo a medical and sign the paperwork after bidding farewell to Bayern's fans in a statement and video posted on Twitter.

"The decision was very hard to make because you and FCB have and always will be an extremely important part of my life," said the 30-year-old, who won eight Bundesliga titles, seven DFB-Pokals and the Champions League in a 17-year career in Bavaria. "Nevertheless, I would like to gain experience at a new club. My destination is Manchester United. I hope you understand my decision. No one can take away the incredible journey we had together."

The midfielders follow on from last week's arrival of Matteo Darmian from Torino and the £21.5m addition of Memphis Depay from PSV Eindhoven earlier in the summer. All four are expected to be included in the squad for the US tour.