Rashford, who joined United as a seven-year-old, emerged from the youth team in 2015 and has since established himself as one of the stars of the Premier League.
The 23-year-old academy product has gone on to score 77 goals in 232 appearances for United, including three in the Premier League and six in the Champions League this season.
Rashford hopes to see out his career as a one-club player, saying: "For me, I never look beyond Manchester United.
"As a kid, I've never looked beyond Manchester United.
"In my mind, it wouldn't sit right for me to be pulling on another shirt.
"I just want to do the best I can for the club whilst I'm here, so hopefully I'm here for the long-term."
Rashford, who was speaking after winning the Sport for Social Change award at Sports Journalists' Association British Sports Awards, added: "All the good things that have happened this year have come from Manchester United giving me an opportunity as a kid. I don't just remember the last year or the last five years.
"I'm talking about when nobody in my family drove and I had to get to training and United had people to come and take me to the training ground and take me back home when I was six, seven, eight, nine.
"The times they put me into accommodation when my mum was struggling at home, when I was 11 and I was there up until 16, 17.
"It is a lot deeper than what people see sometimes. I think people see some of the stuff that has been happening since I've been in the first team but the bond that I have with United is much greater than these last five years and it will forever be greater than probably my actual career because they are the reason I have become a footballer, they have given me the opportunity to express myself and to find out I had talent in football."