It was revealed earlier this week that Cardozo's future was set to be decided in the coming weeks, and he had been linked with a possible switch to the Premier League.
Manchester City and Cardiff were both understood to be keeping tabs on the 30-year-old Paraguay international.
However, Fenerbahce always appeared in pole position for his signature and a deal now looks close.
"The negotiations between the two clubs are going well and the agreement is practically closed. There are only a few details to be resolved," his agent Pedro Aldave is quoted by Turkish website Fanatik.
"His signing could come at any time," he added.