According to Don Balon, who are renowned for making outlandish transfer claims, the Manchester United manager called the Bundesliga champions to discuss potential deals for attacking James Rodriguez and left-back David Alaba.
Mourinho is reportedly eager to add creativity and physicality to his squad after failing to challenge Manchester City for the Premier League title.
Rodriguez – on a two-year loan at Bayern Munich from Real Madrid – is likely to be the more difficult deal for Mourinho to land.
Bayern have an option to convert his loan into a permanent move for £36.7m (€42m).
A new left-back is high on Mourinho’s transfer agenda after muddling through the campaign with Ashley Young filling the position.
Alaba, 25, has established himself as one of the world’s leading left-backs but could seek a fresh challenge after winning seven Bundesliga titles.
Speaking in March, he said: “I am aware that this decision will come to me and I still feel very comfortable here, but I can also imagine taking a different path, taking the next step or looking for a new challenge.
“I am at an age where I have had a lot of experience and where I would like to take the next step in my career.”