Emery, whose side host United in a fourth round FA Cup tie on Friday, said it was "normal" for a manager's job to be on the line when the team was not performing to their potential.
"I was not surprised because this team has big players and can think about winning the Premier League," Emery told reporters. "They can be in the position to win the league, but the reality is they have lost a lot of points this season."
Ole Gunnar Solskjaer was appointed in a caretaker role after Mourinho was sacked and he has overseen a remarkable turnaround to win six successive league games and go level with fifth-placed Arsenal on 44 points, behind only on goal difference.
"Now, they are very dangerous and in a good moment," Emery added. "I don't know why but this team is different and they are playing with more confidence."
(Reporting by Rohith Nair in Bengaluru; Editing by Greg Stutchbury)
