Golovkin, 36, has twice boxed Canelo, 28, and was equally unfortunate not to walk away with a win on both occasions.
The first fight ended a draw while Canelo outpointed Golovkin last year, and talks of another rematch have since surfaced.
However, Golden Boy Promotions president Gomez claimed Triple G isn't interested in a third crack at his Mexican rival, accusing the Kazakh of only wanting 'easy' fights.
Gomez said: “(Looks) like he does not want Bamba de Canelo anymore. Prefers easy (fights) as before. We will see friends.”
The news comes after Oscar De La Hoya, who owns Golden Boy Promotions and promotes Canelo, claimed discussions for the trilogy fight were underway.
The promoter told ESPN that Canelo, who claimed the WBA ‘regular’ super-middleweight title last month after defeating Rocky Fielding, could fight three times in 2019.
De La Hoya said: “Canelo Alvarez is fighting two to three times this year, and fighting the very best. Last night, I had preliminary talks with Golovkin's people.
“So we threw it out there about fighting for a third time.
“Cinco de Mayo is the biggest week in boxing in Las Vegas. So we're starting talks.
“Canelo is open to anything. Canelo is a fighter who wants to fight anybody to be great, and so a third fight is a possibility.
“We're starting discussions, and we'll see where it leads from there.”
Canelo’s victory against Fielding meant he joined elite Mexican boxers such as Julio Cesar Chavez, Marcos Antonio Barrera, Erik Morales and Juan Manuel Marquez as champions to rule in two or more divisions.
The knockout win against the Brit was also Canelo’s first under his 11-fight, £300million deal with sports streaming service DAZN.
Golovkin was reportedly in discussion to sign with DAZN after HBO left the boxing business in late 2018.
The signing of Golovkin would boost the chances of the trilogy being secured.
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