Boro have identified the Scotland international as the man to spearhead a Championship promotion campaign next season after they failed in the play-offs back in May.
It is understood Rovers are willing to do business but will not contemplate selling for less than an eight-figure fee and they would also prefer to sell him to a Premier League club rather than a Championship rival.
That has not scared off Boro, who have already forked out an initial £5.5m this summer to bring Stewart Downing back to Teesside.
Rhodes has scored 73 goals in 134 Championship games since joining Blackburn from Huddersfield for £8m three years ago.
A fee of £12m would make him the second most expensive player in Boro’s history after Brazilian Afonso Alves.