Costa has been key to Chelsea's blistering start to the Premier League season, the Spaniard scoring nine goals to help the London club to a total of 22 points from eight games.
However, he has not featured in Chelsea's last two games - a 2-1 victory at Crystal Palace last weekend and their 6-0 UEFA Champions League rout of Maribor on Tuesday.
Mourinho revealed on Friday that Costa missed the Maribor fixture due to a viral infection that forced him to stay overnight in hospital.
But the former Atletico Madrid man could still be fit for the trip to Old Trafford, although fellow forward Loic Remy will be out for two to three weeks with a groin injury.
"Remy no chance [of playing]," Mourinho said. "Diego has a little chance, the same as [John-Obi] Mikel and Ramires."
Should Costa be unfit, Didier Drogba will likely be asked to fill in after playing over 70 minutes against Maribor following Remy's injury.
The 36-year-old Chelsea legend, who returned to Stamford Bridge in the close-season, has been struggling with ajn ankle injury this term but Mourinho has faith in him.
"He [Drogba] understands the responsibility," Mourinho added. "He understands the need to push himself to different levels to help the team."
Sunday's contest will pit Mourinho against former mentor and United boss Louis van Gaal, with whom the Portuguese worked for three years during the Dutchman's first spell at Barcelona from 1997 to 2000.
Mourinho cites Van Gaal as one of the biggest influences on his career but insists their relationship will mean nothing come kick-off.
"Man United is Man United for me it doesn't matter who is the manager," Mourinho explained.
"It doesn't matter the table it doesn't matter the players. I always look at them independent of these sectors.
"I don't play against Louis, Chelsea plays against Man United, during the match I have no time to look around and to see who the opposing manager is."
