The Daily Express is reporting the Black Cats were hoping to include Danny Graham or Will Buckley in a deal for Rhodes, valued by Blackburn at £12million.
However the transfer embargo placed on Rovers for violating Financial Fair Play rules stipulate that any out-going deals can not involve players as part of any payment.
"They are the terms we are told. It would have to be a straight cash deal for anybody to try and buy him," Rovers manager Gary Bowyer said.
"But it goes back to the owners and if they're not happy it doesn't get done."