Matthäus made no attempt to obscure the identity of the player Bayern are targeting when speaking on Sky Sport Germany.
"I have a player in mind and I have heard that enquiries are already being made. A former Bundesliga player from Leipzig, currently playing for Manchester City, left-back, can also play centrally. His name is Gvardiol."
Real Madrid have also made contact with the 24-year-old Croatian's representatives, with both clubs described as huge admirers of a player who can operate as a left-back or centre-back. Real Madrid face significant defensive reconstruction this summer regardless of who becomes their next manager, while Bayern Munich are seeking a long-term solution on the left side of defence following the expected departures of Alphonso Davies and Hiroki Ito.
Gvardiol joined City from RB Leipzig in 2023 for 90 million euros and has made 121 appearances across all competitions, contributing 13 goals and 10 assists. His campaign this season was cut short in January by a fractured tibia, but he has returned to training and is expected to be available for Croatia at the World Cup.
City are relaxed about the situation. Sporting director Hugo Viana is working on a contract extension for the defender, whose current deal runs until 2028, and Pep Guardiola has made his own view plain.
"I have no info. I would love Gvardiol to stay here. It is not easy to find a player like him, so hopefully he can stay."
The club believe Gvardiol is happy in Manchester and that extension talks will progress smoothly once formally opened. Any sale would require a fee significantly in excess of what City paid three years ago.