The Blues, who finished fourth in the Premier League last season, have already spent more than £125million on new signings Hakim Ziyech, Timo Werner, Chilwell and Silva.
But Lampard is keen to bolster his squad further before the postponed summer transfer window shuts in October and is now turning his attentions to Rice.
The Sun say Lampard wants Chelsea to sign the England international and believes the versatile star can help get the club back in the Premier League title mix.
While the Blues largely impressed in Lampard’s first season in charge, they still finished way off Manchester City and champions Liverpool.
It is claimed Chelsea are closing in on a deal for Kai Havertz and are now plotting a bid for Rice, who enjoyed a fine season in east London despite West Ham’s struggles.
The Hammers value Rice, 21, at around £80m but Chelsea were able to shave off £20m on their deal to sign Chilwell and believe a similar situation could play out in their talks with West Ham.
West Ham boss David Moyes may be forced to sell before he buys this summer and there is limited interest in his squad other than Rice.
If Rice does join Chelsea, he will hope to learn from Brazilian veteran Silva, who joined the Blues on Friday on a free transfer.
The 35-year-old, who had spent almost a decade at PSG before signing with Lampard’s charge, said: ‘I am so happy to be joining Chelsea.
‘I am delighted to be a part of Frank Lampard’s exciting squad. I’m here to challenge for honours and I look forward to playing at Stamford Bridge very soon.’