The Blues have been strongly linked with the England international since the end of last season, with Lampard said to be a keen admirer of the defensive midfielder.
A recent report claimed that the 21-year-old would push for a return to Chelsea if his former club make an offer for his services before the transfer deadline on October 5.
According to The Mirror, the London giants are preparing one final push for Rice, who is believed to be valued by West Ham in the region of £70m.
The Hammers have the midfielder on a contract until the end of the 2023-24 campaign, placing them in a strong position when it comes to any potential negotiations.
Rice was released from the Chelsea academy system at the age of 14 before linking up with West Ham.