Per James Robson of the Evening Standard, manager Ole Gunnar Solskjaer said Pogba would need a "miraculous recovery" from the ankle injury that kept him from participating in the international break with France.
Martial and Shaw will be absent due to muscle problems, while Aaron Wan-Bissaka and Jesse Lingard are doubtful because of a back injury and illness, respectively.
"There's maybe Aaron [Wan-Bissaka], maybe Jesse [Lingard] and I think maybe that's it from the ones that pulled out, but I can't promise those two will be available either.
"Pogba won't be fit unless he's had a miraculous recovery overnight from yesterday to today or tonight to tomorrow. He's not been ready to do 100 per cent training."
He added that Pogba and Martial will likely be available for United's following Premier League match against West Ham United on September 22.
Football writers Liam Canning and Tom McDermott felt that with their injury crisis, the Red Devils are suffering for failing to bring in more reinforcements in the summer:
Pogba is their most creative player and one of their only options capable of unlocking a defence from midfield, so his absence will do the team no favours.
As for Martial, he has thrived in a central role this season with two goals and an assist in the team's first three matches, so putting the ball in the net will fall on Marcus Rashford and Daniel James.
United have not won since beating Chelsea 4-0 in the opening weekend, a match in which Pogba and Martial both contributed a great deal going forward.
The Foxes look set to be among the sides challenging United for their place in the Premier League's top six this season. It will be a difficult afternoon for the Red Devils, doubly so because of the injuries they're carrying.