Perisic, 33, has been one of the key players for Inter this season and his extra-time double won them the Italian Cup against their bitter rivals Juventus.
But the Croat has been linked with a move to Tottenham Hotspur where he would reunite with former Inter manager Antonio Conte.
“It’s a regret, but he wanted experience in the Premier League,” said Inter’s CEO Giuseppe Marotta.
“We were not in a position to be able to present him with a better offer.”
Perisic, who scored eight goals and made six assists this season, was reportedly demanding six million euros ($6.3 million) a season.
Inter are keen, however, to attract the 28-year-old Argentine playmaker Dybala.
“Clearly, the hope is that he can play with us,” said Marotta who admitted, with a smile, that he did not yet know where the “Joya” (the “Jewel” in Spanish) will play next season.
Nor did Marotta rule out a possible return to Inter for Romelu Lukaku. The Belgian striker left for Chelsea last summer for nearly 115 million euros but is unhappy with his situation at Stamford Bridge.
The transfer market does not officially open until July 1 in Italy.