The Birmingham Mail claim four clubs have been offered the 34-year-old ex-Liverpool star, who has 73 caps for Portugal, is a former Champions League winner with Chelsea in 2012 and has been playing most recently for Turkish giants Fenerbahce.
Leicester, Swansea and Watford are also said to have been given the chance to snare Meireles as a free agent signing outside of the transfer windows.
But Albion boss Tony Pulis is unlikely to take the plunge given that his squad is already well stocked with experienced midfielders such as Gareth Barry, Grzegorz Krychowiak, Claudio Yacob, Jake Livermore, while he also has Sam Field coming through.
Meireles made 35 Premier League appearances for Liverpool, who paid FC Porto £11.5 million for him in 2010, lifting the League Cup with the Reds before joining Chelsea for £12m after just one season.
He won a Champions League medal when Chelsea beat Bayern Munich on penalties in the final under the leadership of Roberto Di Matteo, despite missing the game through suspension.