Rodgers, 37, was out of work having being sacked by Reading in December after just six months in charge.
He also had seven months at Watford, before quitting the Hornets to move to the Madejski Stadium.
BBC Sport Wales claim Rodgers has accepted the position and will be unveiled on Thursday or Friday.
The news will come as relief to Scunthorpe supporters, who had feared an approach for the Iron manager Nigel Adkins.
Huw Jenkins, the Swansea chairman, had described Adkins as a target but also conceded he would struggle to pry him away from Glanford Park.
An approach was never made and after failing to get Gary Speed, Billy Reid or Paul Tisdale the Swans have now turned to Rodgers.