Shaw's decision to pull out of United's match against Swansea City through injury on 6 November drew Mourinho's ire, with the Portuguese coach saying that players have to be ready "to do anything" for the team.
The former Chelsea boss revealed before the match that the England international had declared himself unfit for the game at the Liberty Stadium after playing in the Europa League defeat against Fenerbahce on 3 November.
Shaw returned to first-team action in the summer after suffering a double leg fracture in September 2015.
"I was with Luke Shaw at Southampton," Lallana was quoted as saying by Sky Sports.
"He's a very strong boy. He's had a horrific leg injury. You don't come back from that without being mentally and physically strong.
"You don't come back from a leg break like he had without having difficult times mentally on your own."
Shaw reportedly told teammates that he was "baffled" by Mourinho's comments after the Swansea match.
"There is a difference between the brave, who want to be there at any cost, and the ones for who a little pain can make a difference," the former Inter Milan and Real Madrid manager was quoted as saying by the Daily Mail after the game in south Wales.
"If I were to speak with the many great football people of this team, they will say many times they played without being 100 per cent. For the team you have to do anything. That is my way of seeing.
"We have players with problems. To compete you have to go to the limit. It is cultural for some – and this is my culture. More than me, it is Man United."