On-loan from Manchester United, Rashford can be signed permanently for €30m - though Barca hope to renegotiate that price in their favour.
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Other giants, including PSG, are watching the England international closely, but his No1 choice is to join Barca outright sometime in 2026.
"(I want to stay for) as many (seasons) as possible. We will see," he told TNT Sports. "I have to focus on trying to do my job and trying to help the team in whichever way I can."
Rashford added: "I'm learning a lot, it's a new style of football," and "But it's making me a better player. Honestly, it's an amazing experience. I've always been a huge admirer of Barcelona, every player my age has watched them play, it's amazing football."