The Icelandic international is being monitored by Leeds with his contract at Pescara due to end after the Serie B play-off final.
Pescara are on the verge of reaching the Italian top flight but the Italian club have allowed Bjarnason’s deal to run down and the 27-year-old will be available on a free transfer if he opts to move on.
He and United head coach Rosler know each other well having worked together at Norwegian outfit Viking between 2006 and 2009.
Bjarnason – an attack-minded midfielder – began his senior career at Viking and remained there until 2011 when he secured a transfer to Standard Liege in Belgium.
Pescara took him to Italy two years later but Bjarnason spent the 2013-14 season at Sampdoria after signing for them in a co-ownership deal. He returned to Pescara last summer but is likely to seek another transfer if Pescara fail to win promotion.
Pescara and Bologna play the first leg of their Serie B play-off final tonight, with the return leg scheduled for Tuesday evening. Discussions about Bjarnason’s future are expected to intensify once the promotion fight is settled.
The midfielder is an experience Iceland international with almost 40 caps to his name.
He has scored 12 times from midfield for Pescara this season, including a goal in the club’s play-off semi-final win over Vicenza.
Bjarnason’s agent, Jim Solbakken, was asked by the YEP if he was aware of interest from Elland Road. Solbakken said that “on that I cannot comment.”
Leeds recruited heavily from the Italian leagues with limited success last summer and the club will pay closer attention to the British market during the current transfer window.
United are close to signing Scottish striker Lee Erwin on a three-year contract from Motherwell – a move which is likely to be sealed when Erwin returns from a holiday in America – and they have been looking at domestic centre-backs after failing to reach a quick deal to land ex-loanee Sol Bamba on a permanent basis.
Discussions with Bamba – a defender with ample experience of the Championship – are ongoing but the two sides are yet to agree on wages.
A fee with his current club Palermo has already been negotiated.
Rosler, however, admitted on the day of his appointment as head coach that he was happy to look abroad for signings, despite the extent to which a cosmopolitan squad at Leeds struggled in the Championship this season.
Rosler said: “It’s all about balance and if you get the balance in the team right between know-how and hunger, spirit, talent, I think you can go a long, long way.
“For us, we are under financial restrictions so we have to look where we are getting that sort of talent. Sometimes you have to go outside the UK to find that sort of talent. The balance has to be right.”
United owner Massimo Cellino is currently in Italy attending to personal business matters but it is not known if he plans to travel to watch Bjarnason tonight.
Meanwhile, Leeds have agreed a new deal with development squad captain Alex Purver.
The 19-year-old midfielder will take up a one-year contract when he returns for pre-season later this month.
Centre-back Jake Skelton, striker Luke Parkin and left-back Tyler Denton have also accepted new one-year deals.