It is claimed that Wednesday were willing to cover Rhodes' £35,000-a-week wages, however, Boro are only interested in a permanent transfer and if they can sign a replacement.
Rhodes only arrived at the Riverside in a £10 million deal from Blackburn Rovers in February and scored six goals to fire the Teessiders back to the Premier League.
However, the subsequent signing of Alvaro Negredo from Valencia has dented his chances of regular first team football and a move to Hillsborough, where his father, Andy, is employed as goalkeeping coach, has been rumoured for some time.
Wednesday, though, are walking a fine line in regard to the Football League's Financial Fair Play regulations, as hinted by head coach Carlos Carvalhal last week.
Despite this, the Owls could yet land Rhodes' Boro teammate Adam Reach before Wednesday's 11pm deadline, according to BBC Radio Sheffield.
Adam Reach (right)