Bannan arrived at Hillsborough five years ago from Premier League side Crystal Palace.
The Scot has been a fixture under Wednesday boss Garry Monk and has started all four of the club’s Championship matches so far this season.
Sheffield Wednesday’s Championship rivals Brentford have slapped in a bid to lure away the central midfielder to the capital before Friday’s domestic transfer deadline.
But the Owls have knocked back the offer and are keen on keeping the player in Yorkshire as they seek to chip away at their 12-point deduction in the Championship
Bannan, who wears the armband for Wednesday, is in his final year of contract at Hillsborough.
And it remains to be seen whether the Bees will return to the table with another offer for Bannan.
The 30-year-old Scotland international has had spells at Aston Villa, Derby County, Blackpool, Leeds United and Bolton, in addition to Crystal Palace.