Filippo Inzaghi wants to keep on-loan Chelsea midfielder Marco van Ginkel at AC Milan for another season.
Liverpool defender Kolo Toure is close to agreeing an extension to his contract which expires next month.
Howe managed the npower League One side for two years before moving to Turf Moor last January and was immediately linked with the post following Paul Groves' exit last week.
Clarets chief executive Lee Hoos offered a firm rebuttal in the face of such speculation and Howe later stated he was "proud to be Burnley manager" and that "we feel there is a long way to go" with his job in east Lancashire.
But reports attaching the 34-year-old with a second stint at Dean Court have not gone away, with some bookmakers installing him as a 1/6 favourite for the post.
Hoos offered "no comment" on Howe's future following an enquiry from Press Association Sport on Thursday morning, when the club confirmed their manager took training as normal at Burnley's Gawthorpe training base.
The last time a manager left a Championship club to take job in the division below was when Simon Grayson moved to Leeds from Blackpool in 2008.