Odemwingie expected to sign on
Odemwingie was the Baggies' leading scorer with 15 goals last season and has been the subject of speculation linking him with a move away from The Hawthorns, but Hodgson has seen positive signs from Odemwingie since returning for pre-season training that suggest he will be happy to stay.
Hodgson said: "Peter has been excellent in training this week. He sails through the training when everyone else is fighting hard to do the work. He seems to do it with relative ease.
"He seems very satisfied, very happy, so I presume the contract discussions are going okay. Don't forget, he does have a contract with the club. These discussions are to maybe prolong it and reward him for what he did last season.
"It's not as if he is out of contract but I'm confident the chairman and Dan Ashworth [technical director] will reach an agreement with him. I shall work on the basis he is a happy bunny and a West Brom player until that fateful day when someone tells me he isn't."
Hodgson added: "I think we should be pleased our players attract interest. If we are going to improve as a team, we need players in our team that other clubs envy and would like to have. That for me is a sign your club is moving forward, when the players you have got are being envied."
Hodgson believes he is "a maximum of three players" away from having a squad he will be satisfied and is hopeful midfielder Zoltan Gera will return on a Bosman transfer from Fulham.
Gera spent four years at Albion before moving to Craven Cottage during the summer of 2008 and is expected to re-sign once he has recovered from Thursday's ankle operation.
Hodgson said: "I make no secret of the fact Zoltan is a player I would be very interested in bringing back to West Brom. It is where his career began (in the Premier League). I thought he continued it very successfully at Fulham and, now he is looking to move on again, it would be great if we can bring him back to West Brom.
"Talks are ongoing. The operation may have complicated things slightly but I'm still confident that operation will go well and we will be able to sign him."