The 35-year-old has been a stalwart at Hillsborough for the last decade, amassing over 400 appearances and being a crucial cog in the Owls machine.
However, speculation regarding his future has emerged amid continued off-field turbulence in S6.
Wednesday have again had trouble paying players, with pressure on under-fire owner Dejphon Chansiri growing at a rapid pace.
Wrexham were linked with the midfielder earlier this month, but fears were eased somewhat by reports claiming Bannan was keen to stay put.

Barry Bannan eyed by trio
SportsBoom have now claimed Bannan is edging towards the exit door as he grows increasingly disillusioned by the club’s instability.
Derby County have been credited with interest, while Stoke City and Coventry City are also said to be monitoring the veteran’s situation.
He is thought to have agreed a new Owls contract in principle, but it appears he could still depart Hillsborough with the expiry of his current deal fast-approaching.
Sheffield Wednesday in disarray
The loss of such a revered figure would be yet another humiliating blow for a club that has lurched from one disaster to another of late.
Bannan has previously signalled his intention to see out his career at the club, something that would only make his exit harder to take.

Wednesday have been hit by a three-window fee restriction, as well as a registration embargo.
A statement issued by the EFL read: “The EFL can confirm that Sheffield Wednesday Football Club is now subject to a three-window fee restriction after exceeding 30 days of late payments between 1 July 2024 and 30 June 2025.
“In accordance with EFL regulations, the restriction will cover the entirety of this summer’s window and two full windows thereafter (winter 2026 and summer 2026).
“The club has notified the EFL of its intention to appeal this decision and this will be heard in due course.
“Separately, on 3 June 2025, the EFL issued charges against the club and owner Mr Dejphon Chansiri in respect of non-payment of player wages in March 2025 and May 2025.
“This matter is being considered by an independent disciplinary commission, and because of the ongoing proceedings, the league is unable to comment further at this time.”