Meireles, 32, joined Liverpool from Porto back in 2010, but lasted just one season at Anfield before joining Premier League rivals Chelsea in 2011.
Again, the Portuguese managed just one season for the Blues, joining Turkish side Fenerbahce in 2012, where he has been ever since.
The experienced midfielder has been a regular first-team player in Turkey - but may now be looking to leave in order to make one last hurrah in the Premier League.
The Daily Mirror believe that Meireles, who has one year remaining on his contract at Fenerbahce, is desperate to return to English football, and has instructed his agent to make it happen.
The report suggests that agent Jorge Mendes has been told to ask Everton and West Ham United whether they would be interested in signing Meireles this summer.
West Ham may need another midfielder given that Alex Song is likely to head back to Barcelona at the end of his loan deal, though much will depend on who the manager is at Upton Park next season, with out of contract manager Sam Allardyce likely to leave the club.