The club are desperate to push through a deal before flying out to the US on Monday morning, with a representative travelling to the South Coast on Wednesday to enter new discussions over the 25-year-old, according to the Daily Mail.
Southampton have previously rejected a £20million offer - described as "not serious" by Saints boss Ronald Koeman - for the France international who has been linked with Arsenal and Liverpool, and are holding out for around £27m.
The paper reports that an official from United was driven to Southampton in a club vehicle at some point on Wednesday.
Koeman is resigned to losing Schneiderlin, but wants a fair price for the in-demand midfielder. A quick sale would also benefit Southampton as they search for a potential replacement.
“Every player has his price and we have experience as a club of selling players,” Koeman said earlier this week.
“Last season we sold some players, and if there is interest in Morgan then OK. But it has to be at the price the club wants. He is one of the key players or our team, an international player. And I don’t think is a problem for Manchester United.”
Koeman also admitted that he is planning for the new season without the talented Frenchman in his squad.
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“No. I know the situation. I have spoken a lot with Morgan,” Koeman added. “Last season we finally said: ‘Stop all this business because we are not selling any more players’. That was difficult for him but he came back after a few days with good motivation and had a great season. But we know the situation now.
“There is a lot of interest in the player. With a club like United, we know it’s difficult to keep the player. But if they pay, then OK, we can make the transfer.”
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