The Saints parted ways with Carlos Carvalhal after he failed to keep them in the Premier League.
Going for Stam is an odd move by Swansea, given his failure at Reading, but Sky Sports suggest he is a top candidate for the job.
Another man in the frame is the former Wales manager Chris Coleman. Graham Potter of Ostersunds is another candidate.
Swansea will hope to appoint a manager before the end of the month.