Piatek only signed for Genoa in the summer, but has taken Serie A by storm with nine goals in seven appearances, and 13 goals in all competitions.
The Pole put four goals by Lecce in the Coppa Italia, and has scored in every single league game this campaign.
The Blues are in the market for a new striker after problem with the current crop.
Alvaro Morata cost the club £57m last summer in what was a club-record at the time.
However, after an early-season hot streak, his efficiency in front of goal abandoned him and he ended the year on 11 goals in the Premier League.
The Poland international has been touted with a move to several prestigious clubs across Europe but it seems Chelsea will firm up their interest ahead of the mid-season transfer market.
Genoa are understood to be desperate to keep hold of Piatek until the end of the season at least. However, the report also indicates that the 23-year-old could be allowed to depart if a fee worth £40 million is offered.