Defeated Championship play-off finalists Watford were also interested in the Hungarian but Tony Mowbray's Boro appear to have won the race.
Varga, who spent time last season on loan at Greuther Furth, is set to fly to England on Sunday and he could then complete his move in the early days of next week.
Debrecen owner Gabor Szima confirmed to haon.hu: "Jozsef Varga will travel for a medical to Middlesbrough on Sunday and he is expected to sign a one-year loan contract next Tuesday,".
Varga, 25, had already revealed his interest in another loan after admitting on Wednesday he wanted to leave Debrecen.
"I played the last half season in Germany and I would like to play abroad again," he told Debrecen's official website.
"Fortunately my present club are positive and supportive of me. I am happy that there are approaches and interest from other clubs. I think my future may be decided within a week."