The 24-year-old made the move to the Hammers from Swansea City last summer, and he featured on 35 occasions for David Moyes's side during the 2022-23 campaign.
Downes played 21 Premier League matches but found it difficult to secure regular starts, and there will be more competition for midfield positions during the new season.
James Ward-Prowse is believed to be closing in on a move to West Ham from Southampton, and according to Football Insider, Downes could be moving in the opposite direction.
The report claims that Leeds United, Middlesbrough and Crystal Palace are also keen on Downes, but Southampton believe that they are at the head of the queue for his signature.
Downes played under now Saints head coach Russell Martin at Swansea, and it is believed that he favours a move to St Mary's despite the fact that he will have to drop down a division.
Meanwhile, according to the Manchester Evening News, Manchester United have accepted an £7.8m bid from Bayer Leverkusen for goalkeeper Matej Kovar.
The 23-year-old impressed while on loan at Sparta Prague during the 2022-23 campaign, and a number of clubs have allegedly been showing an interest in his services this summer.
Kovar was involved in the early stages of Man United's pre-season, but he did not travel with the club on their pre-season tour of the United States.
A number of teams had allegedly expressed an interest in signing the stopper on loan before Leverkusen placed a £7.8m bid on the table, which has been accepted.
Kovar has been with Man United since 2018, but he has not made a single appearance for the club, spending time out on loan at Swindon Town, Burton Albion and Sparta.
There had been suggestions that the Red Devils would block his exit this summer due to injuries to senior goalkeepers Dean Henderson and Tom Heaton.
Elsewhere, Sheffield United have reportedly come to an agreement with Coventry City to sign Gustavo Hamer.
According to journalist Mike McGrath, Hamer will cost the newly-promoted club £15m, which will be their biggest signing of the summer transfer window thus far.
The 26-year-old's future has been the subject of speculation this summer, with Fulham and Leeds United among the clubs to be credited with an interest.
However, Hamer is now seemingly closing in on a switch to Sheffield United, with personal terms unlikely to be an issue.
The midfielder made the move to Coventry from PEC Zwolle in July 2020, and he has represented the Championship outfit on 132 occasions, scoring 19 goals and registering 24 assists in the process.
Hamer was in excellent form for the Sky Blues during the 2022-23 campaign, scoring 11 goals and registering 10 assists in 45 appearances in all competitions.
Meanwhile, he has played twice this term, including Coventry's Championship opener against Leicester City last weekend.