Griezmann completed his €120 million (£107m/$134m) move to Barca last week after spending five years at Atleti playing under Simeone.
His exit left a sour taste in Madrid, with Atletico adamant they should have been paid €200m (£179m/$225m) because they say Griezmann and his new club agreed terms before July 1, when his release clause dropped by €80m .
Atletico believe the agreement has been in place since March, and made clear their "disgust" with both Barcelona and Griezmann.
Barcelona are confident there is "no evidence" of such an agreement taking place.
"We contacted Griezmann after he made the video saying goodbye to the fans, there was nothing in March," Josep Maria Bartomeu said as the player was introduced to his new side.
And, despite the bad blood between the two clubs, Griezmann has now tried to smooth over the relationship, offering heavy praise for his former coach.
"I am lucky to have learned from Cholo Simeone. Thanks to him, I've been playing at a very high level," Griezmann said in an interview with Barcelona.
"At Atletico, there's never a day off. It's very physical and there's a lot of hard work on tactics. Nobody can let their guard down."
Cross-city rivals Real Madrid emerged victorious in a penalty shoot-out after Griezmann had hit the crossbar with a spot-kick in normal time.
"I know that if I win a Champions League with Barca, it won't make up for the pain of losing that final with Atletico," Griezmann added.
"For the fans, for my team-mates, that was a tough moment."
Barcelona will be hoping he can help them return to the top of the European game.
The Spanish giants last won the Champions League in 2015, when the club topped Juventus 3-1 in the final.