The 24-year-old central midfielder joined West Ham from Swansea City last summer for a transfer fee worth £12m.
Last season, Downes made 21 league appearances for the Hammers, starting only seven of them.
This season, after the departure of Declan Rice, David Moyes is trying to reshape his midfield and the 24-year-old has been strongly linked with an exit.
Southampton boss Russell Martin, who worked with Downes during his time at Swansea City, is keen on a reunion.
It is suggested that Southampton and West Ham discussed Downes as part of the James Ward-Prowse negotiations.
However, it has been claimed that Downes is expected to join the Saints on a season-long loan deal.
Downes has significant experience of the Championship and Martin will be looking to see him hit the ground running on the south coast with Saints.
It remains to be seen if there is any option to buy in the loan agreement between the two clubs.
