Rangers are rooted at the foot of the Premier League without a win all season and the new manager will have plenty to think about ahead of his first meeting with owner Tony Fernandes since replacing Mark Hughes two weeks ago.
The budget, though, may be restricted with Fernandes having already pumped millions in to a largely unsuccessful project.
Former Everton stars Tim Cahill (NY Red Bulls) and Landon Donovan (LA Galaxy) are keen to return to England during the American off-season, but the player top of Redknapp's list is Galaxy striker Keane, who he signed for Tottenham in 2009.
"The MLS market is one that we've got to look at," Redknapp said.
"Robbie's a good shout. He probably would be an interesting player.
"I haven't spoken to him yet no but he would be an interesting player. He's a good pro and he would be the ideal type to have around for sure because he would give us something."
Another more realistic option for Redknapp is Spurs winger David Bentley, whose loan deal with Russian club FC Rostov ends in January.
"Bentley's a good player," the 65-year-old added. "It shows that he must have wanted to play football to go to Russia so credit to him.
"He could have stayed and took his money but he's gone out to Russia. Dave's a good footballer - I haven't had problems with him.
"He was unlucky at Tottenham because he had Aaron Lennon and Gareth Bale in front of him."