Griezmann had been tipped to complete a switch to Barcelona after the World Cup, but has instead elected to sign a new deal with Atleti, keeping him at the Wanda Metropolitano until 2023.
The France star confirmed he would stay in Madrid in a documentary on Movistar Plus, which was produced by Pique's investment company Kosmos.
Following the broadcast, Pique tweeted his thanks to Griezmann, wishing him well for the future at Atleti, and has since been force to defend himself, claiming he had no involvement in Griezmann's decision other than to suggest it be filmed.
However, speaking before Eric Abidal's official unveiling as the club's new sporting director, Barca vice-president Mestre expressed his displeasure with Pique's actions, as well as those of Griezmann's international team-mate Samuel Umtiti, who shared the documentary on his social media channels.
"We will speak with [Pique] and Umtiti internally after the World Cup about what happened but we won't make any more public comments," he said.
"We feel bad for the supporters who have felt disappointed with the way it happened. The president spoke with Pique to tell him that his involvement provoked surprise and discomfort at club and among supporters.
"We respect Griezmann's decision to renew with Atletico as we respect any player's right to choose their future.
"We want committed players who have decided they want to be a success here above anything else."
me faltan las palomitas https://t.co/5nLCaZjuJp— Samuel Umtiti (@samumtiti) June 14, 2018
Meanwhile, Barca could find themselves on the end of an official complaint from Atleti, who have been outraged by the Catalan giants' conduct during the saga.
Barca president Josep Maria Bartomeu confirmed in May he had held talks with Griezmann's representatives, and the documentary affirmed these claims.
"We will decide whether or not to file a complaint against Barcelona for having negotiated with Griezmann," Atleti chief executive office Miguel Angel Gil Marin told Spanish publication Ser Deportivos.
"We have proof of contact. Griezmann is the protagonist with us, in Barcelona he would be a soldier in [Lionel] Messi's service."