It has been reported by the British tabloid that Boro want to pair Rhodes with Patrick Bamford in attack next season.
Bamford featured for the Championship club during the 2014-15 season on loan from Premier League giants Chelsea.
The Sun claims that Middlesbrough want the 21-year-old English forward to return for another spell at the Riverside Stadium.
Rhodes is one of the best strikers in the Championship and was in superb form for Blackburn in the recently concluded 2014-15 campaign.
The 25-year-old Scotland international striker scored 21 goals and created 38 chances in 45 league appearances.
The former Ipswich Town striker had a pass accuracy of 73% and a shot accuracy of 45%, and he won 70 headers, nine dribbles and 12 tackles.
According to a report in The Daily Mail in April, Derby and Forest are interested in the former Huddersfield Town man.