Bayern said on Thursday the 26-year-old, who has been training with them recently, could join the squad until at least the middle of March, with a longer term deal subject to negotiation with Major League Soccer (MLS).
Donovan, who has won over 100 international caps, was the top scorer in the MLS regular season with 20 goals in 25 matches for his Los Angeles Galaxy team.
"Landon has a verbal agreement with MLS to play in Europe for two-and-a-half months, like Beckham's," Klinsmann told the club's website.
"He's made such a good impression here that we've agreed to talk to MLS. The goal will be to see if we can get him long term, at least until the end of the season."
Donovan has spent a week and a half training at Bayern and scored both goals when the first team, minus players away on international duty, beat the club's B team 2-1 in a practice match on Wednesday.
