Inter have agreed a loan move with an option to buy worth £19.6m and have already bought another central midfielder, Roma’s Radja Nainggolan, this summer.
Vecino played under Sarri when he was in charge of Empoli and the 59-year-old would prefer him in midfield over Tiemoue Bakayoko or Danny Drinkwater.
Sarri, who is expected to be busy in the final nine days of the transfer window, believes the 26-year-old can help him stamp his style on the current squad.
The Blues opened talks with Inter over Vecino at the weekend but look set to miss out on defender Mattia Caldara, who is closing in on a move to AC Milan with Juventus team-mate Gonzalo Higuain.
Meanwhile, Chelsea are set to hand Michy Batshuayi a key role this term with Olivier Giroud set to fall down the pecking order if he stays at the club.
Alvaro Morata will remain first-choice striker with Chelsea set to miss out on Higuain.