The former Manchester United midfielder is currently a free agent after the expiry of his contract in June.
West Ham, who signed Lingard on loan during the 2020/21 campaign, have been one of the favourites to land the player on a permanent deal.
While Irons boss David Moyes is determined to complete the deal, talks have been slow due to the 29-year-old demanding a reported wage of £180,000 a week.
With West Ham's lengthy negotiations proving fruitless, Nottingham Forest have pounced and are now leading the race to sign the midfielder.
Last season's play-off winners are understood to be in advanced talks with Lingard's representatives as they look to beat West Ham to the deal.
Forest have been priced at 11/10 to complete the signing, while West Ham have dropped to second-favourites at 11/8.
Should Lingard arrive at the City Ground, it would mark a momentous moment for the Nottingham outfit whose highest earning player last season was on loan forward Keinan Davis who pocketed £45,000 a week.
Other contenders for Lingard's signature include MLS side LA Galaxy. The Californian side are no stranger to high-profile signings, and they could certainly do with a boost considering they sit seventh in the Western Conference.
LA Galaxy are priced at 20/1 to make the deal a reality, while Everton and Newcastle are not far behind on 25/1.
DC United, who have recently appointed former Manchester United star Wayne Rooney as manager, are next on the list at 33/1.
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