The versatile England international has been out since mid-April with an ankle injury but played for United's under-21s on Monday and came through unscathed.
After three consecutive league defeats in which they failed to score, the return of Jones would be a boost for United who are four points clear of fifth-placed Liverpool.
They face Crystal Palace away on Saturday, then Arsenal at home and finish the season at Hull City.
"(As a team) we have got a long way to go as there are still three games coming up, three massive games, and we have got to approach it in the right way. I'm sure we can do that and cement that top-four finish," Jones told the club's website.
"I'm happy to play in the Under-21s and get some fitness. I feel I'm ready to go again and hopefully I can make the squad on Saturday. Hopefully, I'll recover on Tuesday, then train on Wednesday and Thursday, and see how it goes."
(Reporting by Martyn Herman, editing by Ed Osmond)
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