The offer is believed to have triggered a release clause in the Brazil international's contract.
Costa, 24, scored 10 goals in La Liga last season, and 20 in 44 games in all competitions.
Atletico are reportedly prepared to let him go as they have signed strikers David Villa from Barcelona and Leo Baptistao from Rayo Vallecano.
Costa joined Atletico from Valladolid in 2010 and made his international debut for Brazil in March this year.
Costa, a powerful 6'2" striker, would become the fourth most expensive signing in Liverpool's history after Andy Carroll (£35m), Fernando Torres (£26m) and Luis Suarez (£22.8m).
The attempt to sign Costa is not however believed to mean Liverpool are resigned to losing Suarez.
On Thursday boss Brendan Rodgers told LFC TV: "(New signings) 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.
"We're really working hard to bring in the type of profile that can really help us push on."