The 25-year-old joined Boro last summer on a season-long loan from Debrecen.
Despite being signed by Tony Mowbray, Varga was utilised heavily by new boss Aitor Karanka, who played him at right-back for much of the campaign.
Karanka has confirmed he would like Varga to stay and the player admits he hopes to be back on Teesside next season after making a good impression.
"I'd like to stay at Middlesbrough," he told Nemzeti Sport.
"When I left the English club many people said they would like me to go back. I hope we'll meet again.
"I can't say any details except that my future may be decided within a week."
Debrecen previously stated that Boro had until May 20 to take up their option on Varga.