The 25-year-old Italy international will sign a deal worth £45,000 per week and has flown to London to undergo a medical.
Chelsea boss Antonio Conte has been frustrated during most of the summer with regard transfers, most recently losing out on Alex-Oxlade Chamberlain to Liverpool having already losing out on Romelu Lukaku to Manchester United.
Tottenham could also beat their London rivals to the signing of Swansea forward Fernando Llorente but the Premier League champions now have their sights on Zappacosta.
The right-back has four Italy caps and has spent two seasons at Torino, having joined from Atalanta.