The Hornets are in the market for a holding midfielder as they plot a push for promotion next season, and are thought to be ready to rival Boro for Varga.
However, Middlesbrough remain in pole position and are favourites to sign the Hungarian, who spent the second half of last season on loan with Greuther Furth in the Bundesliga.
Debrecen still regard Varga as a key player but have indicated that the highly-rated 25-year-old's future could be decided shortly.
"We are happy that he's our player but in football anything can happen," said Debrecen director Geza Roka.
"It's possible that his future will be decided within a short time. But so far there're no concrete news."