The midfielder was one of 10 players to arrive from the Serie A side last summer shortly after the Hornets' takeover by the Pozzo family, who control both clubs as well as Spanish side Granada.
Abdi has proved a successful addition having made 24 league starts and scored eight goals, the third highest among Watford's ranks behind 19-goal top scorer Matej Vydra, another Udinese loanee who is also in talks to turn his season-long stay into a permanent deal.
Abdi, 26, is giving little away as to the progress of the discussions, but told the Watford Observer: "Yes (talks have started) but I can't say anything about it at the moment. We have talked a little bit.
"Of course I like the football, I like being near London, I like the club and the guys here but I can't say anything because it is too early."
Watford are currently third in the Championship and out of the automatic promotion places on goal difference.