England worldwide Rashford scored twice for United in their 2-1 victory over Liverpool on Saturday, having made his first start since Boxing Day.
This season, the England global has now scored six times in the league, but he has mostly been used as a substitute.
De Boer, who lost his job after five games and 77 days in charge of Crystal Palace, suggested that the pressure on Mourinho to get results is hampering Rashford's progress.
After all the discussion of whether Rashford is fading away under Mourinho at Man United, the Special One threw him a start against Liverpool and he kindly paid him back with a brilliant brace to win the staunch rivalry.
Saturday's victory was the Portuguese manager's 68th Premier League game as United boss and Rashford has played a part in 59 of them.
"Normally he's an English player, you want him given time, he can make mistakes".
Mourinho used his press conference on Tuesday ahead of their Champions League round of 16 clash with Sevilla, to fire back at the Dutchman.
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The Spanish side had the better of the first leg, but a combination of De Gea's saves and poor finishing left them frustrated. Sevilla fell to a demoralizing 2-0 defeat to Valencia at home and will look to bounce back.
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Rashford has similarly been deployed from the bench in Europe, but has three goals in the competition so far and will hope he has done enough to earn a starting place for that game.
"I read a quote from the worst manager in the history of the Premier League, Frank De Boer - seven matches, seven losses, zero goals - that Marcus would not learn under me", said Mourinho. He wants to get results.
'I try to give the best to the kid.
Frank De Boer's words come across as unusual due to the reality of the situation as Jose's love for the youngster has been publicly stated multiple times.
Do you think Rashford deserves more game time under Mourinho? "But if you go to his numbers and how many matches he played with me last season and this, I would say probably he's in the top five players with most matches in the two seasons". "And Marcus is having these experiences".
'I am really pleased with everything. Of course not. He's a player we trust. Not many were expecting him to start against Liverpool and he had a good match not just about scoring goals. "I'm really happy with Marcus".