The Blackburn player is no longer a first team regular at Ewood Park and is expected to be available for a fee in the region of £1.5m.
It is believed that talks are under way, although clubs will also be interested in the former Welsh intrenational.
A move for Savage would make some sense, especially with Dickson Etuhu leaving soon for the African Nations Cup, and Graham Kavanaghexpected to finally leave the club.
That would leave Sunderland's squad depleted in midfield and Keane might welcome the 33-year-old Savage's experience and industry in the Black Cats' relegation dogfight.
Savage has played 18 times for Blackburn this season, his last game was the 3-2 home defeat to Arsenal in the Carling Cup this month.
Keane is warning against expectations of big changes in the transfer window and says he doesn't foresee a wholescale restructuring of the squad.
And he is by no means certain that he will get in all the players he wants.
He said: "We are looking at bringing in new signings because I think there's no doubt that we need to add a bit of quality.
"But I'm not looking at bringing in a lot of players, maybe two or three. It's a difficult time because there are loads and loads of managers out there in the same boat as us.
"Managers are looking for quality players to add to their squad, but there aren't a great many of them around and the competition is intense."
Keane has told the Echo that making full use of the loan system might be the way forward and Jonny Evans and Danny Simpson are two of the players who might be potential captures under that arrangement.
Both were at the Stadium of Light on Wednesday to watch their Manchester United team-mates outgun Sunderland 4-0.
But it's a fact that Keane has been heavily involved in scouting missions this season both at home and abroad.
He has handed his targets in to chairman Niall Quinn and chief executive Peter Walker and now is up to the board to see whether they can secure the signings that are likely to prove so key so Sunderland's season.
Either way, Sunderland need to have a far better second half of the season than they produced in the first.
Only 14 points have been taken from the first 19 games. And though Sunderland are still very much in touch with the teams in the lower half of the table, their manager knows results will have to improve.
He said: "We are all on a learning curve at the club – players, staff, myself. We have to learn fast because this is a very unforgiving league and we've already found out that almost every mistake you make gets punished.
"But I think we have a good enough squad to stay up even without strengthening. We've been unlucky with injuries to key players like Carlos Edwards, Dean Whitehead and Kieran Richardson – players who are really important to us. But hopefully in the new year we'll have all our key players back and we can look forward to a better second half of the season.
"We don't plan to bring many people in during the transfer window but hopefully the ones we do will strengthen us where we've been lacking and help make a big difference to the side."
Sunderland season ticket holders can apply for tickets for the February 2 trip to Liverpool (5.15pm kick-off) from Wednesday until January 15, using voucher 37. Tickets are priced £34 (adults), £25.50 (senior citizens) and £17 (juniors).