Bedimo is set to become a free agent in the coming weeks, with his contract at Lyon due to expire.
The Cameroon international has been offered a new deal by Lyon, but he has refused to commit his future.
Middlesbrough are looking to strengthen their team to ensure that they avoid the drop in their first season back in the top flight.
Aitor Karanka’s side finished in second place in the Championship this past season to earn their promotion.
Boro have already been heavily linked with Dennis Praet and they have completed a deal to sign Viktor Fischer, as they show great ambition in the transfer market.
West Brom, meanwhile, have funds to spend - with Tony Pulis eager to bolster his options.
Bedimo made 22 appearances for Lyon in Ligue 1 and the Champions League this past campaign.