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.
Barcelona's highly-rated midfielder Robert Navarro is reportedly set to join Monaco on a five-year contract.
Japan enjoyed a dream start to their World Cup Group H campaign with a 2-1 win against 10-man Colombia in Saransk.
Huddersfield are set to step up their bid to sign Club Brugge forward Anthony Limbombe with a reported £13million bid mooted.
West Brom have been linked with a shock move for Jeremain Lens, who only joined Besikitas on a permanent basis earlier this month.