Former Borussia Dortmund coach Thomas Tuchel is in talks to succeed Arsene Wenger as Arsenal manager in the summer, according to reports.
Arsenal and Chelsea are both reported to have approached the former Dortmund and Mainz coach.
Bayern chief executive Karl-Heinz Rummenigge, president Uli Hoeness and sporting director Hasan Salihamidzic tried to persuade Tuchel to take over at the Allianz Arena over a conference call, as per German reports.
Current Bayern head coach Heynckes came out of retirement after Carlo Ancelotti was sacked last September and will step down from his role at the end of the season.
The 72-year-old praised Tuchel's qualities and publicly urged Bayern to the former Dortmund coach as his successor.
He said: "Thomas Tuchel has the quality to coach Bayern. I rate Thomas Tuchel. He has continued the progression from the youth teams at Mainz. That is the stairway to success that you must take, learning the correct approach and people skills.
"Under Tuchel, Borussia Dortmund played excellent football with a good system – all of today's modern footballing elements were there for everyone to see. Tuchel's side finished runners-up, won the DFB Pokal and played attractive football. I had fun watching BVB, that's why I respect and rate him as a very good trainer."
Last week, former Bayern midfielder Ze Roberto ruled out the possibility of Heynckes staying on as head coach at the Allianz Arena beyond the summer.
"Recently I had the opportunity to eat with the players and him, but he said to me that he has no desire to continue," the Brazilian was quoted as saying by FourFourTwo.
"He wants to have more time for his wife and family. Had it not been Bayern, he would not have returned anyway. The club means a lot to him."