The Reds manager believes the Brazilian has signed a new deal with Atletico Madrid, all but ending the club's pursuit of the forward, and also quashed a move for Aly Cissokho
Costa has emerged as a transfer target for the Reds this summer but both Atletico and the player himself have poured cold water on a move to Anfield for the Brazilian.
"I believe Costa has signed an extension. There’s nothing in terms of players to add," the Reds boss told reporters.
"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 dismissed a move for Valencia’s Aly Cissokho, following reports that the move broke down because the Liga club were only interested in granting a loan move.
"There’s nothing on the Cissokho story," he added.