Griezmann completed his long-awaited switch to Camp Nou last week and is expected to make his Barca debut on their pre-season tour of Japan.
Barca paid €120m for Griezmann but Atletico argue the fee they received should have been €200m as they claim the Catalan club were in talks with the striker before his release clause fell on July 1.
Griezmann announced his intention to leave Atletico in a social media video posted before the end of the LaLiga season, but the Rojiblancos say his move to Barca had been agreed some weeks earlier.
Reports in Spain have claimed the Royal Spanish Football Federation (RFEF) has opened proceedings against Barcelona over the club's move for Griezmann but, in an interview with Le Parisien, Abidal said the move for the France forward was above board.
He said: "The president [Josep Maria] Bartomeu got in touch with Atletico Madrid to discuss the terms of a transfer for Griezmann and to inform them, if negotiations weren't possible, that Barca would pay his €120m buyout clause. It was €200m in March and €120m in July.
"Atletico are defending their interests and we are defending Barca's. We never had a prior agreement with the player. We got in contact with his lawyer after Griezmann announced he wanted to leave the club.
"We can always talk about interest. We can like a player because our sporting department says they'd fit the profile of what we are looking for. But this interest only became concrete when the player expressed his wish to join Barca.
"He was a very important player for Atletico Madrid. He can make a difference in the big matches and is capable of playing in various positions. He's still a great goalscorer and Barca need players like him."
Despite adding Griezmann to a forward line already containing Lionel Messi, Luis Suarez, Ousmane Dembele, Malcom and Philippe Coutinho, Barca have been linked with making a sensational move for Neymar.
Paris Saint-Germain sporting director Leonardo has confirmed the Brazil forward wants to leave the French capital with a return to Camp Nou two years after his world-record move said to be Neymar's preferred option.
Barca are reported to have offered PSG a choice of six players - including Coutinho, Dembele and Malcom - as makeweights but Abidal suggested the club's work in the transfer market could be complete.
"We needed six attacking players and, with Griezmann, we have those six," he added. "After that, it's the coach [Ernesto Valverde] who makes the decisions."
Messi is reportedly pushing for Neymar's return with the Barca captain said to be underwhelmed by Griezmann's arrival.
"That isn't Messi's role," Abidal added. "If he gives his opinion on someone who players for another team, it's disrespectful to his current team-mates. I believe that's why Leo didn't want to talk about Antoine."