According to Corriere dello Sport, Ibrahimovic, who serves as a senior advisor to RedBird chairman Gerry Cardinale, phoned Modric twice in recent days to reassure him that the club wants him to stay and continue playing a significant role in the squad.
The calls are said to have helped tip the balance in favour of a new deal, with confirmation now only awaiting Modric's signature.
Modric's contract at San Siro had already expired, technically leaving him a free agent this summer, and he had said throughout the process that he would only make a decision once the World Cup was over. Milan's hierarchy, including owner Cardinale and new head coach Ruben Amorim, had already been working for weeks to convince the Croatian to commit to another year before Ibrahimovic became involved.
Amorim has also taken a personal interest in retaining Modric, saying at his introductory press conference that he had spoken to the veteran twice already and would continue to do so.
"He's a player we want to keep," Amorim said. "I'm not saying he'll play every match, but we want to count on Modric next season."
The head coach added that he believed Modric would rejoin the squad in the coming days after resting following the World Cup.
Modric made 37 appearances for Milan last season, scoring two goals and providing three assists, and has become an important dressing-room presence despite his advancing years. Should the extension be finalised as expected, he is set to link up with the squad during their pre-season tour of Australia, which begins on July 26.
Amorim is reportedly planning a more carefully managed role for Modric this season, mindful that he cannot be expected to shoulder the same workload as last year at his age.
Alongside his contribution on the pitch, the Croatian is increasingly seen as a mentor figure for Milan's younger midfielders, including Samuele Ricci and Ardon Jashari, with his experience viewed as valuable both physically and in the dressing room as the club looks to improve on last season's fifth-place Serie A finish.