The set date placing the Rossoneri and the Austrian Football Association in a simultaneous race for the 67-year-old's answer as the World Cup looms.
Rangnick departed for North America with Austria's squad without resolving either situation. La Gazzetta dello Sport reports that he has been impressed by what he heard during a six-hour meeting with Milan owner Gerry Cardinale and is seriously considering the proposal.
A further discussion with Cardinale, Zlatan Ibrahimović and chief executive Massimo Calvelli is scheduled for this week before Rangnick gives both parties his verdict.
AC Milan are seeking a sporting director to lead their total reconstruction after the dismissal of head coach Massimiliano Allegri, sporting director Igli Tare, technical director Geoffrey Moncada and CEO Giorgio Furlani — all of whom departed the day after the club's fifth-place Serie A finish confirmed a second successive season without Champions League football.
Rangnick has made clear he would not join as head coach. He wants to lead the club's football operations from a directorial position, and has drawn up a list of trusted collaborators he would require. First among them is Johannes Spors, currently Southampton's sporting director, who would serve as the primary football appointment.
Rangnick also wants Christopher Vivell — currently Manchester United's director of recruitment — as head of scouting, alongside other longtime associates from his Red Bull network.
The Austrian Football Association has responded by placing a two-year contract renewal worth approximately €2m net per season on the table, and federation president Josef Pröll has acknowledged publicly that the situation is now in Rangnick's hands.
"He still needs time," Pröll told APA.
Austria open their World Cup campaign against Jordan on June 17. The federation would like to know their coach's intentions well before that date.
Milan, regardless of the Rangnick outcome, are also progressing separately on a head coaching appointment. Oliver Glasner remains the preferred candidate of the club hierarchy following the six-hour meeting last week, and Cardinale has continued contact with the Austrian's representatives. Mauricio Pochettino remains an alternative option.