Lampard, who enjoyed a trophy-laden 13-year spell at Stamford Bridge as a player, is seen as the ideal man to develop the club’s young talent.
The Chelsea legend led Derby County to the Championship play-off final in his maiden season in the dugout before losing to Aston Villa at Wembley.
Sarri’s future will be resolved this week after winning the Europa League crown in style by hammering Arsenal in Baku.
The Italian, however, is at loggerheads with Blues bosses over transfer deals and that could drive him out of the club.
He reportedly told the club on Friday he wanted to return to Italy with Juventus his next stop.
The Serie A champions are on the hunt for a replacement for Massimiliano Allegri.
Sarri, who ran the Old Lady close as Napoli boss, is among the favourites for the job.
Chelsea are willing to release him from his contract if Juventus meet his £5m compensation fee.
The Europa League winners have refused to comment on rumours but Starsport understands Sarri would stay if he had more control over the comings and going of players.
He was left frustrated when Cesc Fabregas was allowed to leave in January with no replacement lined up.
Despite mounting criticism from fans and pundits, Sarri led the club to third place in the Premier League and lifted the Europa League title – his first piece of silverware.
Chelsea were close to sacking him earlier in the season but are prepared to let him carry on if he wants to.
But if he accepts the Juventus vacancy then Chelsea will turn to Lampard as his successor.
Chelsea are facing a transfer window ban spanning two windows imposed by FIFA.
They have appealed against the decision but would have to rely on their loan army and academy talents to beef up their first-team squad if they lose.
And Chelsea bosses believes Lampard is the only candidate to oversee such a project.