The 23-year-old was a fixture in the Championship for the Swans last season, but his future at the club is under the scanner amid talk the Welsh side will cash in.
Downes has attracted strong interest from the Premier League in the ongoing window and Swansea are open to letting him go for a £12m fee.
Top flight outfit Crystal Palace are keen to take the midfielder to Selhurst Park and are ready to table a proposal of £9m for his services.
But Swansea are yet to agree a deal with the Eagles for Downes as they want a fee closer to their valuation of the player.
And the Hammers have now joined the race for Downes’ signature and are willing to match the offer Crystal Palace have tabled for him.
Swansea are aware that selling Downes would help to balance the books and he has in the London duo.
It remains to be seen which Premier League club the Swans star ends up playing for next season as he is highly likely to leave the Welsh giants this summer.
