The People says Rodgers will make an extraordinary SOS call to Drogba in a bid to sign the former Chelsea striker on a short-term deal.
Drogba, 34, could be available to join a club in Europe as soon as the Chinese season ends next week. His club Shanghai Shenhua are believed to be willing to rip up his contract, making Drogba a free agent and releasing him for a move back to the Premier League, on the understanding that he is welcome back in China for their new season.
If all goes well Drogba also has an option to sign a longer-term agreement at Anfield.
Rodgers knows Drogba from his time as a Chelsea coach and is desperate for strikers, with Luis Suarez his only fit senior frontman.