According to Calciomercato.com, the Blues and Manchester United have joined Serie A clubs AC Milan, Inter Milan and Juventus in the race to sign the teenager, but Genoa president Enrico Preziosi will "not entertain any offers less than €40 [million to] 45 million."
Additionally, Preziosi is said to hope a bidding war will drive Pellegri's price up. Milan are believed to be looking to get ahead of the game by moving for him in the coming weeks.
The Italy under-17 international has caught the eye in Serie A this season, most notably when he netted twice against Lazio in September.
His agent, Giuseppe Riso, recently talked him up as boasting characteristics like a young Zlatan Ibrahimovic, per Squawka News:
The young forward certainly boasts impressive finishing technique as well as power, while he is a physically imposing presence in the penalty area.
The Blues could do with a new forward option given that manager Antonio Conte does not seem to trust back-up option Michy Batshuayi, per Sky Sports' Peter Smith.
Pellegri has had little time to show how he can perform on a consistent basis.
But it is likely he would relish working under former Juventus and Italy manager Conte, while he would likely be happy to play a back-up role to Alvaro Morata at Stamford Bridge.
Chelsea will need to be certain of Pellegri's quality, though, if they are to splash out a sizeable sum for a player so young.