The 22-year-old Slovenian striker, who joined from RB Leipzig in August, is expected to miss about four weeks of action, with his return targeted for December.
Sesko was hurt during the 2-2 Premier League draw against Tottenham on November 8, coming off as a second-half substitute in the 82nd minute.
Initial and follow-up scans have confirmed that the injury is not major, playing down earlier fears from the club's fan base.
He was ruled out of Slovenia's World Cup qualifiers and is now set for a period of rehabilitation.
Sources indicate he will be out for "about a month at most," though the precise length is still to be determined.
Head coach Ruben Amorim is scheduled to provide an update this week ahead of the team's first post-international break fixture.
Sesko is set to miss the matches against Everton and Crystal Palace.
United then face a congested six-game December fixture list - against West Ham, Wolves, Bournemouth, Aston Villa, Newcastle, and Wolves again - which the club hopes Sesko will be able to rejoin before or during.
Since his move, the 1.95m forward has made 12 appearances for the Red Devils, scoring two goals.
Out-of-sorts Germany international striker Niclas Fullkrug is desperate to leave West Ham in the January transfer window, according to reports, but not to another Premier League club amid reported interest from Fulham.
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Matthijs de Ligt's resurgence at club level has propelled him back into the Netherlands national team, almost a year after his previous appearance.
Bournemouth general manager Tim Bezbatchenko has claimed that the club will not stand in star forward Antoine Semenyo's way if a big club makes an offer for his services.
