Of the three, Wednesday are the only ones to be credited with interest in the 31-year-old this summer, with the Owls seeing an offer worth an initial £100,000 turned down by Blackburn, according to Sky Sports.
Charlie Mulgrew in action for Celtic
It is little surprise, therefore, to see Carlos Carvalhal's side installed as 2/1 favourites for Mulgrew's signature - although with odds of 4/1 and 6/1 respectively, Leeds and Celtic could perhaps give Wednesday a run for their money.
The pair are also low on numbers in Mulgrew's preferred role of centre-back, with Leeds forced to play young midfielder Conor Shaughnessy there in recent matches, while Celtic are currently without Erik Sviatchenko, having failed to replace the released former Premier League star, Kolo Toure.
Mulgrew - who spent spent six years at Celtic and captained the Bhoys before leaving last summer - was relegated with Blackburn last season, but admitted earlier recently he would be open to a return to the Championship, where Wednesday and Leeds continue to compete.
See the full list of contenders below:
Sheffield Wednesday - 2/1
Leeds United - 4/1
Celtic - 6/1
Bristol City - 8/1
Brentford, Cardiff, Hull, Preston, Sheffield United, Sunderland - all 12/1