Blackburn Rovers face a testing final week of the winter market.
The Daily Star Sunday says Swansea City have discussed a £11million deal, which would see Rhodes replace £28m Manchester City signing Wifried Bony, 26.
And there could be renewed interest from Hull City as manager Steve Bruce hunts for a striker.
The Tigers had a £10m deal rejected in the summer.
It is also understood that neighbours Burnley are considering Rhodes, 24, as an eventual replacement for Danny Ings, 22, who is expected to leave the Clarets in the summer when his contract ends.
