Costa was not considered fit enough to start the 2-1 win over Stoke which sent Chelsea seven points clear.
The striker was introduced at half-time but lasted just 11 minutes as his fragile hamstrings forced his withdrawal, with scans to determine the full extent of his injury to take place early next week.
Mourinho would not be drawn on an exact prognosis, but suggested Costa will miss the April 18 clash with third-placed United, which is followed eight days later by a trip to second-placed Arsenal.
"It's too early," Mourinho said.
"He has to go home, he has to wait 48 hours
He has to do all the scans again.
"He will be out for sure a couple of weeks, but we have seven weeks until the end of the season.
"Seven weeks and eight matches
For sure, in half of it he has to be with us."
The Blues responded from conceding a Charlie Adam goal from his own half and losing Costa to beat Stoke and move seven points clear in the title race.
Stoke midfielder Adam's 65-yard strike soared over Thibaut Courtois to cancel out Eden Hazard's penalty opener before Loic Remy struck the winner after Costa had come and gone.
Mourinho insisted his half-time introduction was not a risk, as Costa was fit to be included.
"I don't say a gamble because we did every test, we did every scan," the Blues boss added.
"The player trained two days 100 per cent with the team, the medical department was convinced that he was ready.
"The player was convinced he was ready, not for 90 minutes, but he was ready to play and to help the team.
"Sometimes as a manager I have to risk things
Medical departments have to do the same, if they want to be top medical departments like our department is.
"The safe medical department cannot work with me
People with fear, afraid to risk, cannot work with me.
"My medical department is fantastic, they do fantastically well for us.
"Today between them and the player and myself, things didn't happen the best way, but I'm happy with that, because it's football."
Mourinho's side lead second-placed Arsenal by seven points, with United a further point behind in third, while holders Manchester City are nine points behind in fourth ahead of Monday's trip to Crystal Palace.
The Gunners moved to within four points with a comprehensive defeat of Liverpool, but Mourinho insisted Chelsea's focus is on themselves.
"Pressure (is) because we need to win six matches and we know that," Mourinho said.
"We're not doing our countdown based on the points the second, the third, the fourth lose.
"Our countdown must be made of our victories
At lunchtime (we needed) six victories and one draw, at dinner time five victories and one draw."
Mourinho described Adam's effort as "probably the best goal of the season in the Premier League" and absolved Courtois of blame
The goalkeeper's positioning was fine, according to Mourinho, who instead lamented an earlier foul on Hazard and Chelsea's failure to press for the ball.
"I didn't enjoy, but it's a goal every top player in the world would love to score," Mourinho said.
"It's a fantastic goal, but we made a mistake
It's a clear foul on Hazard, but we cannot stop waiting for the referee to give (it).
"He (Courtois) was in the correct position, (but) when you lose possession of the ball you have to immediately react and to adapt your position
The goalkeeper has to adapt his position, the players have to press the ball.
"Our reaction was to look for the referee and to wait for a foul."
Stoke boss Mark Hughes praised the audacity of Adam, but rued missing out on the Potters' first Stamford Bridge point in 21 years.
"It was an outrageous attempt and thankfully it's gone in," Hughes said.
"At the time I wasn't quite sure what he was attempting, but it became very clear.
"Fantastic vision and technical ability to beat a goalkeeper of Courtois' quality from that distance."
Stoke are 10th after a third straight loss, which came after an "uncharacteristic" Asmir Begovic error, when the goalkeeper rolled the ball out poorly, leading to Remy's winner.
"On the day he had a fantastic performance," Hughes added.
"That was his one mistake, and Chelsea had the ability and talent to capitalise on it.
"When they get in a winning position, it's usually very difficult to stop them."
Source : PA
Source: PA