Rovers have confirmed their interest in the 34-year-old who is set to be a free agent when leaving boyhood club Middlesbrough.
Tony Mowbray is keen to add the experienced winger to his ranks for next season and hopes his close relationship with Downing could convince him to make the move to Ewood Park.
The former England international is unlikely to be short of interest however, with Sheffield Wednesday and Rangers already linked, while there is expected to be overseas clubs eyeing a possible move.
His Middlesbrough contract expires at the end of this month and as things stand he will be leaving the Riverside for pastures new.
The Teesside Gazette report Rangers and Blackburn Rovers are both targeting Downing this summer with Sunderland the latest club to register an interest.
A host of other clubs in England are monitoring the former England midfielders situation plus there is long-standing interest from the MLS.
Boro are currently without a manager following the end of season departure of Tony Pulis and while Downing could yet have a future at the club if a new deal was offered, at this stage that looks increasingly unlikely and he is set to depart as things stand.
The Gazette report: “The player is expected to discuss options with Rovers and Rangers when he returns from his end of season holiday later this month.
“Sunderland may appeal as a destination because it will enable to stay in his native North East and cause the minimum disruption to his young family.”
Downing had a loan spell at Sunderland in the 2003-04 season. He made seven first team appearances for Mick McCarthys side in the Championship.