The Blues have already spent over £80m signing Hakim Ziyech and Timo Werner from Ajax and RB Leipzig, respectively.
Those deals were funded by the departure of Eden Hazard to Real Madrid last summer.
Lampard wants four more new faces through the door before the new Premier League season gets underway on September 12.
Bayer Leverkusen’s versatile midfielder Kai Havertz has told his bosses that he wants to move to Stamford Bridge and talks are moving forwards behind the scenes.
Defenders Ben Chilwell and Declan Rice, of Leicester and West Ham, are also high on Lampard’s wishlist.
As is a new goalkeeper as the Chelsea boss has lost faith in Kepa Arrizabalaga, who joined from Athletic Bilbao for £71.6m in 2018.
If Lampard gets all of his top targets then the net spend will amount to around £300m.
And the Athletic claims Abramovich is happy to spend such a sum after UEFA relaxed their Financial Fair Play regulations this year due to the coronavirus pandemic.
However, the Russian will not shift from his philosophy that the club should be self-sustainable.
That means Lampard will have to decide which players to move on and Granovskaia will be tasked with finding them a new home – which may be difficult in the current climate.
One agent who represents Premier League players told the Athletic how difficult it will be to sell stars in this window.
“The market at the moment is really difficult,” they said. “It’s a buyer’s market, that’s for sure.
“For clubs, players and everyone else associated with the industry, fantasy football is over. This is a global economic crisis that has affected football in the same way as every other business in the world, and it wasn’t planned for.
“Who would have said in February that we’d be looking at no income through gate receipts, commercial sponsorship, hospitality and so on?
“Clubs are now looking at their playing staff and saying, ‘We can’t afford to carry deadwood’.”