Mourinho hit out after Smalling and Shaw missed United's 3-1 win at Swansea City on Sunday.
Mourinho hit out after Smalling and Shaw missed United's 3-1 win at Swansea City on Sunday, making a distinction between "the brave" and "the ones for whom a little pain can make a difference".
But Southgate, who left both players out of his squad for England's games against Scotland and Spain, said the defenders would not have missed the United game unless they felt unable to play.
"They didn't play, so there's a medical issue," he said. "Chris hasn't played for the last four games.
"We have really good relationships, medical-to-medical, with all clubs. I've got great trust in our medical team that they'll make the right calls.
"Obviously it's a difficult one because I don't know the reason for the comments Jose's made."
Smalling has missed United's last four games with a foot problem, while Shaw has endured niggling injuries since making his comeback from a double leg fracture sustained in September 2015.
On Shaw, Southgate said: "If he wasn't fit to play yesterday (Sunday) -- and he was with us in September under (former England manager) Sam (Allardyce) and was feeling problems with the leg -- there's clearly something.
"After an injury as severe as that, we've got to handle that with care as well. It's a difficult balance to find. But that's where we have to trust our medical teams.
"I think it's so difficult with any player. Every injury is different, every individual is different.
"I've not known players not play unless there's something. Having been a player, that would be my view on it."
England tackle old rivals Scotland in a World Cup qualifier at Wembley on Friday before bringing the curtain down on 2016 with a home friendly against Spain the following Tuesday.
Southgate, placed in temporary charge after Allardyce was brought down by a newspaper sting, was a member of the England team that overcame Scotland 2-0 in a group-stage match at Euro 96.
The game at Wembley was illuminated by a famous goal from Paul 'Gazza' Gascoigne, who lifted the ball over Scotland centre-back Colin Hendry's head before volleying home.
Gascoigne, England's midfield talisman at the time, was a bundle of nervous energy and Southgate revealed the lengths that were taken to calm him down before kick-off.
"(England coach) Bryan Robson made him a fishing rod and he was pretend fishing in the bath at Wembley," Southgate told journalists at England's St George's Park base in Burton-on-Trent, central England.
"Because that was the only time he ever relaxed. Make of that what you will!
"They created (a fishing rod) out of the medical skip. To calm him down, basically.
"He used to go with (goalkeeper) David Seaman during the week in the afternoon just to get himself out of everybody's hair for a couple of hours. He found that the most relaxing thing."
While the England-Scotland fixture no longer resonates as widely as it once did, Southgate is eager to impress the historical significance of the rivalry upon his players.
"It's the oldest international fixture. The rivalry is obvious, the history between the two countries is obvious," he said.
"We could build it up to be as big as we want. For us it's a game of football that is going to help us qualify for a World Cup. That is the key.
"But we should embrace the emotion of the occasion. That's what sport is about.
"Because every time you play for England, you have a chance to make some history, or to play in a game that people will remember forever, and that's incredibly powerful."