The 26-year-old Villarreal forward is in "very advanced" discussions with the MLS Cup holders, agent Vicente Montes has revealed.
Spanish news outlet El Periodico Mediterraneo reported earlier this week that Dos Santos is set to leave Villarreal this summer following a disappointing season with the La Liga club that saw him record just six goals and seven assists in 40 total appearances.
According to the newspaper, the Mexico international has entered discussions with the MLS Cup holders over a potential €7 million [£4.9 million] switch to LA ahead of the league’s transfer window opening next month, having turned down an offer from Liga MX’s Tigres to return to his homeland.
Agent Vicente Montes has now confirmed those reports and added that the California-based outfit have been chasing his client’s signature for some time, telling Univision: "It's a strong possibility, very strong, because LA has long-been interested in signing Gio.
“He has had a very intense life in a very short time and is contemplating the different stances that are being presented.”
Montes also noted that no other clubs have made a legitimate approach for Dos Santos at present, despite interest, and that the former Barcelona and Tottenham Hotspur playmaker was very keen on the Galaxy's proposal.
“That of the Galaxy is one that appeals a lot because he has always shown a lot of interest in them,” he continued. “It is a possibility that we are looking at because the Galaxy is making an effort in hiring him.”
However, if the Galaxy do hope to secure Dos Santos’ high-priced signature in the near future, it could likely only happen following the league’s anticipated introduction of the long-rumoured “core player rule".
The club currently have all three Designated Player slots filled by captain Robbie Keane, Omar Gonzalez and the incoming Steven Gerrard, but the proposed rule would allow teams to add a fourth signing above the league’s maximum wage restriction of $436,250 [£277,637.24].
Still aged just 26, Dos Santos is entering the prime of his career and, as a regular starter for the Mexico national team, would be a major commercial draw were he to make the move across the Atlantic.
With Gerrard eligible to play for the Galaxy from July 8, head coach Bruce Arena does already have an abundance of attacking riches at his disposal but, should the El Tri star also come on board this summer, then the club would no doubt have one of strongest line-ups ever assembled in MLS.