Lampard's stint at Chelsea has seen his side ship 63 goals in 43 matches under the former Blues midfielder, with the capital outfit boasting just two wins from five Premier League outings in the 2020-21 season.
The Blues kicked off their Champions League campaign with a drab 0-0 draw against Sevilla on Tuesday evening, and the Chelsea hierarchy are said to be making preparations for life without Lampard in the near future.
According to Italian outlet Quotidano Sportivo, Roman Abramovich has personally earmarked Allegri as his first-choice replacement for Lampard, and the Chelsea owner is said to have no time for loyalties if he believes that Lampard is not taking the club in the right direction.
Allegri has been out of management since he stepped down as Juve boss at the end of the 2018-19 season, where he guided the Old Lady to five straight Serie A crowns between 2015 and 2019.
Chelsea now face a crunch match against Manchester United in the Premier League this weekend before continuing their European venture against FC Krasnodar next Wednesday.