The forward, along with Alejandro Garnacho, was left out of their squad for the derby against Manchester City on Sunday.
United beat City 2-1 at the Etihad Stadium, their first win there since 2021.
“A move for Marcus (Rashford) would suit the player – a new challenge for him,” Keane said.
“He’s been there a long time. When you’re at a big club and your attitude is not right – and it hasn’t been great recently – sometimes it’s good to part ways.”
On Rashford’s omission by boss Ruben Amorim, he added: “It is a big call. He (Amorim) has obviously seen something he doesn't like.
“But for those who have watched Manchester United recently, it's not a big surprise because both players have been poor. He has said 'enough is enough'.
“It's a big call, but I like to see it - he has put down a marker.”
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