The France international's deal is running down this season and negotiations had appeared off after stalling at the end of last season.
However, La Gazzetta dello Sport says Milan management have now made a final pitch to the goalkeeper.
Milan are offering Maignan a deal worth €7m-a-year, which would put him on par with Rafael Leao as the highest paid at the club.
The agreement would run to 2029 with an option for another 12 months.
For his part, Maigan has been seeking €7.5m-a-year to re-sign.