The Portuguese coach extended his perfect record in European finals to five titles by guiding Roma to the inaugural Europa Conference League trophy.
The 24-year-old England international striker's nine goals have been pivotal in Roma's passage to next Wednesday's inaugural European Conference League final against Dutch side Feyenoord in Tirana.
Inter Milan -- fourth last season -- suffered an early upset after Domenico Berardi slotted in a 27th-minute penalty to give all three points to their rivals from Modena.
Football Transfers: French Ligue 1 side Bordeaux block Malcom move to AS Roma after late Barcelona bid
The 21-year-old had been expected to arrive in Rome on Monday night for a medical on Tuesday before joining his future teammates on their pre-season tour of the United States.
The 29-year-old midfielder Javier Pastore signed a five-year contract with AS Roma that runs until June 30, 2023.
Spanish veteran defender Ivan Marcano has signed for Serie A giants Roma, the Italian side confirmed on Thursday.
Coric made his senior debut for Dinamo in 2014, and went on to win four Croatian League titles and three Croatian Cup trophies there.
Totti now has a director's role with the Italian club, who this season reached the Champions League semi-finals for the first time in 24 years.
The Paris-born Algeria forward is likely to command a fee of around 50 million pounds, with Roma's offer believed to be some way short of that figure.
Edin Dzeko scored twice to help Roma to a 4-1 victory at AC Milan on Sunday and keep the capital side's title chances mathematically alive.