The 24-year-old bagged nine goals in 30 games last season, but he struggled to hold down a starting spot in the first-team.
That has led to talk the Italian, who has won six caps for the national team, could leave this summer and clubs from around Europe are keeping tabs on him.
One of those is Everton and, according to Premium Sport, they have now made an offer of £25.8m.
The deal is structured so that the Toffees would hand over £21.5m upfront, with a further £4.3m paid out via bonuses.
Napoli are reluctant to part with Gabbiadini this late in the transfer window, but they may shift their stance if they can land a replacement.