Barca confirmed the signing of Griezmann on Friday on a five-year deal after a solicitor representing the forward deposited money for his buy-out clause at La Liga headquarters in Madrid.
The signing ends an extended saga involving the Frenchman and Barcelona, as Griezmann previously denied the club last summer to remain with Atletico Madrid.
That decision lasted just one year, as Griezmann announced before the conclusion of the La Liga campaign that he would leave the club at the end of the season before completing his move to Barcelona on Friday.
However, Atletico believe the France international agreed to terms with Barca before his clause dropped from €200m (£180m/$226m) on July 1 and therefore claim they are owed an additional €80m (£72m/$90m) to ratify the transfer, they said in a statement.
Cerezo, the club president, said the La Liga outfit had proof that Griezmann and Barcelona negotiated early and planned to request a greater fee.
"If the club have made this statement it's because there's evidence," he told RAC1.
"We have to study it and we'll demand the amount we consider to be necessary."
Barca, who reportedly plan to present Griezmann at Camp Nou on Sunday, have yet to respond to Atletico's claims.
Cerezo also hit out at Griezmann, who confirmed in May he would be leaving Atletico.
"I'm disappointed by his attitude," he said.
"He wanted to leave, but he didn't tell us until the last minute."
Griezmann, a member of France's World Cup team last summer, originally joined Atletico Madrid from Real Sociedad in 2014 as he went on to make 257 appearances for the club.
He scored 133 goals in those 257 appearances as he departs Atletico Madrid as the club's fifth-highest goalscorer, behind only Luis Aragones, Adrian Escudero, Paco Campos and Jose Eulogio Garate.
During his time with the club, Griezmann claimed a Europa League title, a Spanish Super Cup and a UEFA Super Cup while finishing third in the Ballon d'Or voting two different times.