The Blues are expected to part company with Conte this summer following a largely disappointing campaign which ended with them missing out on Champions League football by finishing fifth in the Premier League.
Talks over a severance package for Conte are still ongoing, with the Italian understood to be holding out for up to £17.5m in compensation should he be sacked.
Sky Sports News reports that Chelsea have nonetheless come to terms with Sarri, but cannot agree an official deal as the 59-year-old is still under contract at Napoli, where he will be replaced by Carlo Ancelotti.
Chelsea would be forced to pay a £7m release clause if they are to appoint Sarri before May 31, so are waiting until next month to wrap up the deal.
Sarri, who spent three years in charge of Napoli, is also wanted by Russian outfit Zenit St Petersburg.