The Reds have confirmed the exit of their 31 Premier League goal striker of last term, with Barcelona set to pay somewhere between £60m-£80m for his signature.
Liverpool have been heavily linked with a move to snap up Wilfried Bony from Swansea as a replacement, but with his value set at £19m, Remy is seen as a good option.
The French striker starred for Newcastle on loan from QPR last season, and has been heavily linked with Arsenal in recent months.
But the Gunners are thought to be dragging their heels with a potential move, and may well pull out altogether after beating Liverpool to Alexis Sanchez.
Rodgers is eager to beat the Londoners to Remy, whose release clause stands at just £9m.
The 27-year-old is also wanted on a permanent basis by Newcastle, but the Merseysiders can offer him a more lucrative contract and Champions League football.
Liverpool may still pursue Bony as well, and are closing in on deals for Lazar Markovic, Dejan Lovren and Divock Origi.