Cameroon international Eto’o is a free agent after being released by Chelsea at the end of last season.
While the Reds have not made an approach for him so far, he is one of a number of options being looked at, sources have told ESPN FC.
The 33-year-old striker has reportedly been in talks over a switch to Ajax as he considers his next move.
Suarez, last season’s 31-goal top scorer, was sold to Barcelona for 75 million pounds in July, and the Reds looked to have sorted out an eight million-pound deal for QPR forward Loic Remy soon afterwards.
Remy got as far as flying to the U.S, where Liverpool were on a preseason tour, to undergo a medical, only for the deal to fall through.
Rodgers brought in Rickie Lambert from Southampton for four million pounds at the start of the summer to assist Daniel Sturridge in attack, while Fabio Borini remains on the books as he is reluctant to leave for Sunderland, who have agreed a 14 million-pound deal with the Reds to sign him.
But the manager feels it is necessary to add another striker to ensure his squad has the strength in depth to cope with the demands of the Premier League and the Champions League.
Eto’o, a former Barcelona and Inter Milan player, arrived at Chelsea from Russian club Anzhi Makhachkala in August 2013.
He scored nine goals in 21 Premier League appearances under Jose Mourinho last season before being released.