The Leicester City assistant boss has ties to Sunderland's new owner.
Sunderland have reportedly turned to Scotland for their new manager with Jack Ross in the frame, as is Shrewsbury Town's Paul Hurst.
Bookmakers are already offering odds on the next Leeds United manager.
The Shrewsbury Town boss is now sixth favourite to succeed Mick McCarthy at Portman Road.
The Shrewsbury Town midfielder has been linked with moves to Norwich City, Leeds United and Sheffield United.
Peterborough manager Grant McCann latest to be considered by Barnsley - what about Bristol Rovers, Charlton and Shrewsbury bosses as well?
Carvalhal's fate as Sheffield Wednesday manager could be decided after Saturday's visit of Middlesbrough.
Reports claim Carvalhal could be sacked by Sheffield Wednesday.
Paul Hurst is reportedly interested in taking over at Sunderland after remarkably guiding Shrewsbury Town to the League One summit.
Shrewsbury Town boss Paul Hurst has been linked with the Sunderland managerial job.