King's injury means the defender cannot play regularly and, in his absence, Jonathan Woodgate wears the armband. But Keane has been linked with possibly taking over the role and Redknapp said: "Jonathan Woodgate is a terrific player as well so we'll have a look at it and make a decision."
The manager could not hide his delight at re-signing the striker, adding: "We're pleased to have Robbie here; I think he'll be a terrific signing for us. He's a good player, great character."
Keane went to Liverpool last summer to join his boyhood idols, but scored only seven goals and struggled to hold down a regular place.
"These things happen, people make decisions in life and move on," said Redknapp. "For whatever reason it didn't work out, but we're pleased to have him back.
"The chairman tried very hard to keep him in the first place and he's done a terrific job and worked hard to get him back, and Jermain Defoe and Pascal Chimbonda.
"All the signings have been made to improve the squad; Cudicini and Palacios are terrific signings, so I am very pleased with the work we've down.
"Liverpool are such a strong side, going well in the League and it was difficult for him to break into the team on a regular basis but he's back here now and he'll be very important to us."
News that Defoe could be out for 10 weeks after undergoing surgery on a broken bone in his foot, Spurs acted quickly for a replacement. "We knew a day or two ago how bad Jermain's injury looked and that is a blow to us.
"He started so well and got three goals and was sure to score plenty more goals between now and the end of the season, so we needed somebody else in and in Robbie we couldn't have got anybody better."