Defeat at home to Liverpool on Sunday confirmed Leicester's relegation back to the Championship.
Keane said of his former United teammate on Sky Sports: “He's a pretty honest guy and no doubt he's disappointed, but his message there was he wants to get them back up next season.
“He obviously wants backing from the board and to get on with the recruitment, the pre-season and to start looking at plans ahead.
“But he's kind of looking for a thumbs up from the club owner and maybe there'll be some sort of statement over the next few weeks and they'll say, ‘Yeah, we're backing Ruud to get the club back up next year’, because it can happen.”
Board problems
Keane added: “There's obviously frustration - the lack of goals at home, the supporters, definitely problems at board level.
“Ruud mentioned recruitment in January - it didn't quite happen - so there's stuff to fix and it’s now down to whether Ruud is the man to do it.
“It's been a tough few months for him, but I think he's handled himself well. Fingers crossed he'll get that opportunity.”