The 34-year-old is without a club at the moment after he left Turkish giants Fenerbahce in the summer after a four-year stay.
Meireles is hoping to a Premier League club in the near future and Leicester boss Craig Shakespeare is hoping to move quickly.
Leicester City's move for Adrien Silva has continued to be blocked by Fifa and Meireles could be a short-term measure for the former Premier League champions.
They face competition from Leicester City, Swansea, West Brom, Watford and Cagliari have also shown an interest.
He would need to prove his fitness quickly with Leicester's schedule set to pick up with just one more international break left before Christmas.