Rashford's future appears to be in the balance after the 27-year-old, a product of United's youth system, was left out of the Manchester derby.
United won 2-1 at Manchester City on Sunday, with Rashford telling journalist Henry Winter on Tuesday that he was "ready for a new challenge and the next steps".
"I don't mind what he said. I think it's spot on," former United captain Keane told the Stick to Football podcast.
"Sometimes a deal just works for everybody. It's been going on for a year or two now, there's obviously a bit of talent in there, he's been at the club since he was a kid.
"A change would probably do him the world of good, for him and his team and his family. Just go and play, maybe go abroad or whatever it might be."
Rashford's current contract at Old Trafford runs until 2028 and although he has been linked to several other clubs, United manager Ruben Amorim is keen for the England forward to stay.
But Keane said: "I always think if you lose that hunger, it is hard to get it back.
"Forget about the few bob (pounds) he's earned over the years -- and listen, good for him --- but we know, the great players, it's nothing to do with that."
Keane added that leaving United could revive Rashford's "real hunger" for on-field success.
"I think there's a good chance he might get it back with a change of a club and scenery, a new country, a new league, whatever, but I don't think he'll get it back at United."
Amorim, talking to reporters ahead of Thursday's League Cup quarter-final at Tottenham, urged Rashford to "speak with the manager" rather than air his grievances publicly.
Rashford has scored 138 goals in 426 appearances for the club since making his senior debut in 2016.
However, he has lost form over the last 18 months with United and was also axed from England's Euro 2024 squad.
Rashford has scored just 15 goals in 67 appearances since signing a lucrative new contract 18 months ago, with speculation over his future increasing in recent days.
The forward, who was absent from training on Monday through illness, is set to miss the League Cup quarter-final at Spurs.
Amorim is keen to keep Rashford but he admitted he would have handled the situation differently when he was a player.
"If this was me, probably I will speak with the manager," he told reporters on Wednesday. "But, guys, let's focus on Tottenham. Tottenham is the most important thing."
© 2024 AFP