Manchester United manager Louis van Gaal has refused to rule out the possibility of signing goalkeeper Victor Valdes.
Manchester City boss Manuel Pellegrini says he is confident that England goalkeeper Joe Hart will sign a new contract.
Keane has been one of Major League Soccer's top scorers since moving to southern California from Tottenham in August 2011.
He has contributed 34 goals and 21 assists in just 55 MLS appearances while helping the Galaxy to win back-to-back MLS titles in 2011 and 2012.
The 33-year-old has also been a member of MLS' Top XI in each of the past two years.
The Galaxy also signed Landon Donovan to a lengthy contract extension last year, while their third designated player, defender Omar Gonzalez, is contracted for the next three seasons.
Galaxy will kick off their new MLS campaign on Saturday against Real Salt Lake.