Suarez said he believed he had more to offer, but claimed he understood Barca's decision to let him leave after six seasons and 198 goals in 283 appearances for the Blaugrana.
However, he also left an indirect message for the board as he said: "When the club tells you they aren't counting on you, you have to accept it, just like when a player tells you he wants to leave."
Messi told Barca in August that he wanted to leave the club and the Argentine believed he was entitled to depart for free due to a clause in his contract.
Barca, though, insisted the clause had expired and told Messi he would only be allowed to move if another club paid his €700 million release clause.
Otherwise, he would have faced a bitter legal battle and that ultimately saw the six-time Ballon d'Or winner decide to stay – at least until his contract expires in 2021.
"It has been a crazy month," Suarez said. "Things have been invented, other things leaked which aren't true. Sometimes that infuriates you, but you have to step away from it.
"When you arrive at Barca, they say 'careful with Leo, with the forwards...' and now it seems as if the striker is bad for Leo. We try to do our best."
Suarez and Messi are close friends and the Uruguayan claimed he had not asked the Argentine for advice ahead of his move to Atletico.
— FC Barcelona (@FCBarcelona)September 24, 2020
"We know each other well," he said. "Leo knows what I think, I know what he thinks. We are quite grown up to be giving each other advice."
Now, the two will be rivals in LaLiga and Suarez said: "Obviously, that will be a strange thing. But we'll see what happens in future.
"I have played against him in Uruguay-Argentina games. It won't change the relationship or the feelings we have."