Donovan, who has had loan spells with English Premier League club Everton and Germany's Bayern Munich in the past, will become a free agent in December if he does not extend his current deal with the Galaxy.
"Right now I am enjoying myself, I want to continue enjoying myself, we have received offers from teams in different countries that are intriguing," Donovan told Fox Sports' Fox Soccer Daily on Monday.
"I am still going to speak with the Galaxy and see how they value me and see where they want to go, going forward, but I am really open in a way that I haven't been before in the past.
"I was pretty content to stay in LA. But if it doesn't work out and we can't come to an agreement here then I am ready for a different challenge and we will see what happens," he added.
Donovan took an extended break from the game after securing the MLS Cup title with the Galaxy in December and lost his place in the U.S. national team.
However, he returned to the international stage at the recent CONCACAF Gold Cup and was the tournament's joint top scorer with five goals, helping Juergen Klinsmann's team to the title.
Losing Donovan would be a blow for the Galaxy after David Beckham left the club last season for a brief pre-retirement spell in France with Paris Saint Germain.
Irish striker Robbie Keane remains as the team's top international alongside Donovan but the club's promise to bring in a big-name player to replace Beckham has yet to materialise.
While Brazilian Kaka of Real Madrid and England midfielder Frank Lampard of Chelsea were both strongly linked with a switch to L.A, the club instead filled their remaining elite 'designated player' slots last week by signing homegrown defender Omar Gonzalez to a new contract.
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