Both Premier League clubs have been linked with the World Cup winner in recent weeks.
And according to Bild journalist Christian Falk, Bayern are willing to sell the 25-year-old.
The Bavarians would be willing to accept a fee of £40m in order to raise funds to sign Manchester City winger Leroy Sane.
Tolisso would represent a much cheaper option for United than Jack Grealish and James Maddison.
Newcastle are hoping to sign the Frenchman as they prepare for new era under their prospective Saudi owners.