But if Mick McCarthy wants to sign the talented Welsh international he will have to shell out a transfer fee because the Baggies are adamant that Koumas will not leave on a season-long loan.
Wigan last week tried and failed to take the 25-year-old on a one-year loan deal.
And Albion came out after that to say that while they would consider offers, the club had no intention of letting him return once he did leave, nor of having him playing against the Baggies while still being contracted to the club in the long-term.
If Sunderland do pursue an interest, then Albion will look for a fee in the region of £1-1.5million for a player who used to be his club's star player until falling from grace.
Koumas is a gifted player with great vision in the centre of the park and is always a danger from free kicks.
The last time Albion were relegated from the Premiership it was Koumas who was head and shoulders his club's star man.
The former Tranmere player stayed with the Midlands club though and helped them bounce back to the top flight at the first attempt.
Since then though he has fallen out of favour with the club.
He returned to pre-season training last summer almost a stone overweight, much to manager Gary Megson's fury and he has not impressed succeeding boss Bryan Robson either.
A fresh start would undoubtedly be beneficial to a player who has lost his way recently.
And McCarthy and his management team could be the group to bring the best out of a player who is a top quality Premiership player when he is firing on all cylinders.
But Sunderland are loathe to take on players whose off pitch discipline can be questioned.
Both Sean Thornton and Jeff Whitley were shown the door this summer amid rumours that their discipline was not up to standard.
McCarthy worked hard to get the best out of Thornton -another who reported back to pre-season overweight last year.
And it will be interesting to see whether he feels he can get the best out of Koumas in the way that he could not out of Thornton.