The Sun says Hammers chiefs held a board meeting last night and opted for the former Chelsea striker ahead of fellow Italian Roberto Donadoni.
Zola could be in place as the club's first foreign boss in time for Saturday's Premier League clash at West Brom.
A West Ham spokesman said: "We've had a detailed and positive discussion and expect to make an announcement very shortly."
Zola, 42, is currently joint-coach of Italy Under-21s.
He may bring his pal and ex-Chelsea team-mate Pierluigi Casiraghi with him if he agrees to take over from Alan Curbishley on a deal worth around £1.5million a year.
West Ham chief executive Scott Duxbury flew to Rome on Sunday for a second round of talks with Zola, after the club finally dropped its interest in Croatia boss Slaven Bilic.