The 27-year-old Ghana international had an impressive World Cup with the Black Stars and is set for a return to the Stadium of Light.
Last season Mensah made 16 appearances in the Premier League for the Black Cats and could play an important part in their upcoming campaign.
It is understood Lyon are keen to offload Mensah from their wage bill, although the two clubs are not thought to have agreed a financial package as yet.
Sunderland manager Steve Bruce told The Journal: "We're close with John Mensah.
"He wants to come and Lyon want him to go, so that will hopefully get sorted in the next few days."
Meanwhile, Bruce is still hoping to land England goalkeeper James, although admits the player's situation is far from clear.
The 39-year-old shot-stopper has been in talks with SPL side Celtic but having cancelled a trip to Parkhead on Monday, a switch to Sunderland could still be on the cards.
Bruce added: "David James has talked to Celtic, which we always knew he was going to do.
"I don't think he is trying to play one off against the other, he's just keeping his options open at the moment. We're still hopeful something will happen."