Ribery, Robben sign one-year Bayern extensions
German magazines Sport Bild and Kicker claim the pair, who were both out of contract in June, have already agreed terms with the Bundesliga champions.
Last weekend, Bayern chairman Karl-Heinz Rummenigge hinted Robben, 34, and Ribery, 35, could expect to be offered extensions.
An official announcement is expected to be made in the next two weeks.
Bayern reportedly received enquiries from clubs in China, Qatar, Turkey and the USA interested in signing either Robben or Ribery.
They have been told to fight for their places next season with wingers Kingsley Coman, 21, currently injured, and 22-year-old Serge Gnabry, who will return from a loan spell at Hoffenheim.
Ribery and Robben both played in Bayern’s goalless draw at home to Sevilla on Wednesday when they reached the semi-finals of the Champions League for the seventh time in nine seasons.
They have played in each campaign and were key figures when Bayern won the 2013 treble of Champions League, Bundesliga and German Cup titles.
Robben scored the winning goal when Bayern beat Borussia Dortmund 2-1 in the 2013 final at Wembley to win their fifth European Cup title.
Ribery joined from Marseille in 2007, before Robben signed two years later from Real Madrid and between them they have made nearly 700 appearances for the Bavarians, scoring 255 goals.