Falcao remains his top target as Rodgers looks to replace the 31 goals lost through the sale of Luis Suarez to Barcelona, although Chelsea and Manchester United are also keeping tabs. But Cavani is a decent back-up option should Falcao fall through.
It could go down to the wire on September 2 as Rodgers admits it has not been easy finding the right players to bring in this summer.
World class addition needed
"The money is there - and we will look to improve at the top end of the field until the very end of the window," Rodgers said.
"It's not easy to get a top level striker, no. But we have to keep looking. We certainly need something, because we haven't brought in another striker so far, apart from Rickie Lambert.
"You just never know what will happen in that last week - the window will always throw up something. We will persevere, and the boys are working hard behind the scenes to see what information we can get on the availability of players."
Initial approach turned down
Despite an initial rejection in their approach for the £50m-rated Falcao, Liverpool are still pursuing the Monaco striker. The Colombian international looks back to full fitness following knee surgery, and is reportedly available at the right price as he's unhappy in Ligue 1. Real Madrid have also been linked.
Liverpool know it will be difficult to get Falcao to commit, so intend to offer him the option of a season-long loan with a permanent buy-out clause attached. That structured deal could also appeal to Manchester United and Chelsea, who are monitoring the situation. Both Chelsea and United have money to spend, United especially, although a striker is not a pressing priority for either.
If Falcao snubs that style of proposal, Cavani is next on the agenda. Like Falcao, the Uruguayan international made a big-money move to France last summer but has not completely settled.
If the deadline nears with Liverpool no closer to landing the world-class forward they require, they may be forced to move for Samuel Eto'o as a stop gap until the Janaury transfer window.