The latest offer from the English club is a deal worth around £30m (€33.5m): £25m plus £4-5m in bonuses.
Napoli want €40m for Allan, 29, but it is felt that a compromise could be struck at €35m.
Everton’s latest proposal puts them in touching distance of an agreement with Napoli for the Brazil international.
Helping Everton’s cause is the fact that Allan has ruled out other options and has his heart set on joining Carlo Ancelotti’s team.
Talks continue with optimism growing that Everton will soon have one of their main summer targets.