England international Rashford limped off in the 61st minute of United's 2-0 loss, with Solskjaer forced to play midfielder Jesse Lingard in attack as he was without Anthony Martial (thigh) and Mason Greenwood (tonsillitis).
Solskjaer confirmed Rashford will have a scan on Monday and he will be hoping for good news given the dearth of attacking options at his disposal after the departures of Romelu Lukaku and Alexis Sanchez to Inter.
"He felt his groin," Solskjaer told MUTV.
"We'll do a scan tomorrow to see how badly he's injured. I don't know how long he'll be out, I'm not a doctor. But he'll probably be out for a little while."
United faces Rochdale in the Carabao Cup on Thursday (AEST) then has three games in seven days against Arsenal, AZ and Newcastle United.
Solskjaer said: "Hopefully Mason and Anthony will be back for the Arsenal game and at least one of them for Rochdale."
The defeat at West Ham left Solskjaer's side with just eight points from six Premier League games.