Amid speculation of Suarez's possible departure, manager Brendan Rodgers has been in the market for another forward, and the offer for Costa is believed to have triggered the player's release clause.
The Brazilian striker netted 10 times in La Liga last year, after arriving at Atletico from Real Vallladolid in 2010.
The move for Costa has been received as something of a surprise, as the forward has yet to make a name for himself in Europe.
Liverpool have declined to comment on the transfer, but Atletico are thought to be happy to allow Costa to move on, due to the arrival of David Villa from Barcelona.
News of the offer follows the Rodgers' comments about future transfers earlier this week.
Rodgers said: "They have to be of the right quality, that's the bottom line.
"We've got money to spend on getting that quality in, but if it's not available I won't waste it for the sake of bringing a player in."
If the Costa deal is successful he follow the likes of Iago Aspas in making the transition from Spain to the Premier League.
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