Griezmann joined the LaLiga champions in a long-anticipated €120million transfer ahead of the 2019-20 season.
Barca were also heavily linked with trying to bring Neymar back to Camp Nou but Messi insists a desire to be reunited with his former team-mate did not mean he was cold on the Griezmann deal.
"Obviously it is a lie that I did not want Griezmann," he told RAC 1.
"Last year I said he was one of the best and the best are always welcome. I also wanted Neymar to come for the same reason, because he is different - one of the best, and to have him here would have been important for our goals.
"But with Ney we had to look beyond sports, many parties did not want him to come [due to] everything that had happened."
Griezmann has scored three times in seven LaLiga appearances but struggled to hit the heights that made him one of the most sought-after players in the world while at Atletico Madrid.
He completed 90 minutes and failed to score in the three games in which Barca have dropped points in LaLiga this season, while he is without a Champions League goal and was an unused substitute for last weekend's 4-0 win over Sevilla.
The France international said Messi's early season calf injury and the fact neither player talks too much had made it "difficult" for them to establish a relationship.
Messi maintains there are no issues and knows adapting to Barcelona and the current interpretation of their famed style takes time.
"For any player who does not know the Barca method, coming from outside is very difficult," he added.
"Griezmann is an intelligent player, of great quality, and he will end up adapting."