The Daily Star claim both Premier League clubs are hoping to land the 31-year-old, with Wolves also said to be interested in the Ivory Coast international.
Bamba only joined the Championship outfit in the summer following a successful loan spell from Palermo.
But following the appointment of Steve Evans, Bamba has been in and out of the starting XI as he battles for competition with Liam Cooper and Giuseppe Bellusci.
Evans is reportedly keen to keep hold of Bamba, but Massimo Cellino is willing to let him move on.
Blackburn Rovers manager Paul Lambert is urging the club not to sell star striker Jordan Rhodes - with Middlesbrough among several prospective suitors.
Boro are in the market for a high-profile attacking addition, but have so far been thwarted in their quest to sign Fulham’s Ross McCormack.
Ex-Huddersfield forward Rhodes remains on their radar, with Boro head coach Aitor Karanka keen to land a forward and another utility man capable of playing in defence and midfield following Jack Stephens’ return to Southampton - with Benfica’s Victor Lindelof being an option despite the move currently being on hold.
Boro have already brought in Athletic Bilbao attacker Kike Sola and winger Gaston Ramirez, but remain keen on McCormack or Rhodes - if a deal can be brokered.
Rhodes has been on Boro’s wanted list for some, with Boro seeing a number of offers turned down in the close season, despite the forward submitting a transfer request in an attempt to force through a move to Teesside.
Despite Blackburn already bringing in loan striker Danny Graham this month, with fellow frontman Fode Koita set to join from Turkish side Kasimpasa, Lambert is keen to keep Rhodes.
He said: “It’s the same situation as it was when I first came to the club. I’ve not heard anything. I chat with the chief executive and until someone tells me different then it’s the same situation.”
Ex-Boro forward Lee Tomlin, linked with Leeds United, has joined Bristol City on a season-long loan from Bournemouth.
Tomlin moved to the Cherries from Teesside for £3.5m last summer.
Sheffield United remain keen on bringing in Fulham defender Dan Burn and are being tipped to make another bid, while young defender Terry Kennedy could leave on loan to boost his game-time levels.
Sheffield Wednesday still have work to do to finalise a full-time transfer with Daniel Pudil, with the deal having encountered a hitch.
The Czech Republic international is on a season-long loan at the club from Watford, with the Owls wanting to sign him permanently for a knock-down fee.
Huddersfield Town remain in the market for winger Rajiv Van la Parra, with their hopes potentially taking an upturn, thanks to Wolves’ likely purchase of Cardiff’s Joe Mason.
Town have made a second bid for Van la Parra and Wolves may be more receptive to sell if they land Mason, with their Dutch winger offering attacking cover as it stands, with the Midlands club in the market for forwards after selling Benik Afobe to Bournemouth.