Messi was hellbent on staying at Barca and was prepared to take a big pay cut to remain at the club before securing a move to Paris Saint-Germain.
But Pique told Laporta that 'nothing bad would happen' if they let the six-time Ballon d'Or winner go.
Lluis Mascaro, director of Spanish newspaper Sport, told Radioestadio Noche: "Messi knows that Pique told Laporta that if Messi left [the club] nothing [bad] would happen."
Mascaro's colleague Toni Juanmarti added: "I do not know where or when, but surely the [number] '3' (Pique) will want to give his version [of the story]."
Messi recently spoke to Sport and touched on a number of different topics, including Laporta's claim that he had hoped the 34-year-old would have played for Barca for FREE.
On Laporta's claim, he said: "The truth is that, as I explained when I left, I did everything possible to stay.
"I was never asked to play for free. I was asked to pay my salary with fifty percent. I did that without any problem.
"That's how I wanted to help Barcelona. It was my and my family's desire to stay in Barcelona. The president's words were really not okay. They hurt me.
"It felt like Laporta took the ball from me but then didn't know what to do with it. He just didn't have to say those things. Now people may think I've caused doubt. I don't deserve that."
Without Messi's services, Barca are struggling as they languish in ninth place in the La Liga table.
They've already lost three league games and trail leaders Real Sociedad by nine points.
The club have also relieved Ronald Koeman of his duties, with legendary midfielder Xavi poised to replace him.
Chosen for YouChosen for You
Most Read StoriesMost Read