Alvarez, 25, stopped Briton Amir Khan in the sixth round of their Las Vegas bout on 7 May to retain the title.
The WBC had given Alvarez a deadline of 22 May to begin negotiations with WBA and IBF champion Golovkin or face being stripped of his title.
But Alvarez says he will now target a fight against the unbeaten Kazakhstani.
"I'm hopeful that by putting aside this ticking clock, the two teams can now negotiate this fight," said Alvarez.
Alvarez, whose only professional defeat came against Floyd Mayweather in 2013, added that he had instructed his promoter to "finalise a deal as quickly as possible".
Golovkin, 34, who now holds three of the four major middleweight crowns, has won 32 of his 35 fights by knockout in an undefeated career, and was invited into the ring after Alvarez's win over Khan.
"I will fight 'GGG' and I will beat 'GGG' but I will not be forced into the ring by artificial deadlines," said Alvarez.
WBA and IBF middleweight champion Gennady Golovkin has been handed the WBC title after Mexican Saul 'Canelo' Alvarez vacated the belt.