HITC Sport understands Sheffield Wednesday are one of the clubs contacted, with Aston Villa and Norwich also possible options.
Rhodes signed for Boro on the last day of the transfer window for £11 million but that only came after they failed to land Karanka's number one target Ross McCormack.
Middlesbrough's Jordan Rhodes celebrates their second goal
Rhodes scored six goals to help Boro to promotion but he was anything but guaranteed a starting spot and he was far from clinical, indeed for the crucial final game of the season with Brighton - he had to do with a place on the bench.
Whilst Boro will be happy for him to compete for a place next season, Sheffield Wednesday have been alerted to his situation by a third party and could try and tempt the Teesside club with an offer.
The South Yorkshire side missed out in the play-offs but wealthy owner Dejphon Chansiri is ready to back coach Carlos Carvajal to go one better and Rhodes is in their sights.
Rhodes himself wants to stay with Boro to have his first crack at the Premier League, but should he leave The Riverside then Hillsborough is likely to appeal given his dad Andy is goalkeeping coach at the club.
Relegated pair Aston Villa and Norwich City are also showing an interest.