Rashford, 22, had drawn the interest of Barcelona in 2018 and 2019.
He was among the four strikers shortlisted by Barcelona as a possible replacement for Luis Suarez at Camp Nou.
“I had a chat with Marcus when he was playing under Jose Mourinho, and he did not have a good time,” Evra told Sky Sports.
“A lot of clubs were coming and asking him to leave.
“I remember he said, ‘Patrice, I will never leave Manchester United because it is the best club in the world’.
“When I heard that as a United fan, I was so impressed and so happy.”
Rashford has been one of Man United manager, Ole Gunnar Solskjaer’s top performers so far this campaign.
The England international has scored 19 goals in 31 appearances for United.
His current deal at Old Trafford will expire in June 2023.