Boss Sarri has grown frustrated by Alvaro Morata's continued struggles up front, while World Cup winner Olivier Giroud has been goalless for the Blues since May.
The Mail say Argentina frontman Icardi is Sarri's dream option.
But the 25-year-old would command a huge fee, even though a £97.5million release clause only becomes active next summer.
In contrast, Piatek would be far cheaper despite being an instant success during his debut season in Serie A.
Former Cracova ace Piatek, 23, already has nine goals for Genoa, including four against Lecce in the Copa Italia.
And after making his Poland debut in a 1-1 draw with the Republic of Ireland in September, he netted the opener in a 3-2 Uefa Nations League defeat at home to Portugal this month.
Sarri is keen to act in the next transfer window as his only real selection difficulties since his summer switch from Napoli hvve been in attack.
He retained Spain forward Morata in the summer – but without reward.
And now at least one new striker seems certain to sign in January.
AC Milan's Patrick Cutrone, 20, who has one cap for Italy, is also a possibility.
And Icardi would certainly be out on his own in terms of cost.
So if Sarri really is determined to get him it could be the first big test of how much Chelsea owner Roman Abramovich will back his Italian manager in mid-season.
So far Sarri has impressed everyone at Stamford Bridge.
His squad quickly adapted to a fast-flowing style that has left the unbeaten Blues third, two points behind Liverpool and champions Manchester City.
MAURO ICARDI and
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