The 25-year-old, who can also play in defence, finished last season on loan at Greuther Furth, who were relegated from the Bundesliga.
Now a number of clubs, including Austrian giants Rapid Vienna, are looking at taking him from Debrecen but Boro may compete for his signature.
Tony Mowbray is in the market for a holding midfielder and Varga - who was a previous target for Swansea City - could be the answer.
"There's a real interest from England and I hope that Jozsef Varga can take this chance," his agent Vladan Filipovics revealed.
"There are a number of clubs from England, and elsewhere," he added to Sky Sports.
Geza Roka, a club director at Debrecen, stated that they were yet to receive an official approach for Varga, who is expected to leave the club this summer.