La Gazzetta dello Sport is reporting that the two parties have intensified contact and the 31-year-old has signalled his willingness to move.
Goretzka's contract at Bayern Munich expires on 30 June. Bayern confirmed earlier this year that they would not offer a renewal, ending a seven-year association with the German champions.
According to the Gazzetta, Milan are offering Goretzka a three-year contract worth €5 million per season — a significant salary reduction from his current earnings at the Allianz Arena but a deal that would make him one of the club's highest earners.
A formal commitment from the player is expected before the World Cup, with Goretzka expected to represent Germany in North America this summer.
The deal is complicated by competition from multiple clubs. Juventus have also tabled a proposal, as have Napoli and Inter Milan from within Serie A. Fenerbahçe and Atletico Madrid are among the clubs pursuing him from outside Italy.
Goretzka has represented Germany 59 times and remains a key figure in Julian Nagelsmann's plans for the tournament. Milan's position as third in the Serie A table, on course for Champions League football next season, gives them a competitive offer alongside their financial terms.
No formal agreement has been reached. The final decision is expected to be announced shortly after the season concludes.