Liverpool reportedly tabled a bid to trigger Costa's release clause at Atletico earlier this month as Rodgers looks to bolster his squad before the close of the transfer window.
Costa claimed he was happy at Atletico Madrid and Rodgers has now indicated that the Brazil international has signed a new deal at the Vicente Calderon to end Liverpool's hopes of luring him to Anfield.
"I believe Costa has signed an extension. There's nothing in terms of players to add," Rodgers is quoted as saying by the Daily Mail.
"We've signed players early on and we hope to sign some more before the end of the window. In the meantime, there's nothing at the minute."
Rodgers also played down talk that the club were close to signing Valencia left-back Aly Cissokho after the Frenchman claimed he had agreed personal terms to join Liverpool.
"There's nothing on the Cissokho story," added Rodgers.