According to the Evening Standard, the Serie A side are huge admirers of the former Napoli boss and are ready to make a move if Chelsea sack him.
Former Leicester City and Fulham boss Ranieri replaced axed Eusebio Di Francesco earlier this month, but only on a short term deal.
Reports further say that Roma president James Pallotta wants a big name to be in charge next season and Sarri is on top of the list.
There were rumours last month that Sarri had already met Pallotta and Roma advisor Franco Baldini over a possible deal, but the 60-year-old Italian dismissed the reports telling Sky Sports: “I have a contract with Chelsea for the next two seasons, so it is impossible to have another contract. I read in an Italian newspaper something about a meeting with Baldini but I think for the moment Franco is in South Africa, so it is very difficult.”