Golovkin's manager Oleg Hermann said: "As a next opponent, Gennady wants Murray.
"We have already made an offer to Martin. The only problem is his unresolved issues with a US visa. If he will deal with this and would be willing to meet with Gennady, we are happy."
It is understood that the fight date had been agreed with US broadcasters HBO.
Murray, 30, lost a contentious points decision to Sergio Martinez in Argentina back in April - the only defeat of his career, while Golovkin's murderous punching power halted Brit Matthew Macklin's challenge in the third round on the weekend.
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