The Mirror says sources close to the Baggies claim negotiations are "progressing nicely" with the 29-year-old who has been a key figure in the club establishing themselves in the Barclays Premier League.
West Brom boss Steve Clarke is reluctant to make any bold claims publicly on Olsson's future. But he did say: "Jonas has a year left on his contract and he understands that. I think he'll make decisions based on what's best for him and his family. Hopefully it's a positive decision for us."
Albion have not ruled out keeping Olsson for the final year of his present contract if talks break down.