The 29-year-old Spanish international has been linked with Liverpool, Manchester United and other European sides as he enters the final year of his contract.
He has told the German champions that he does not wish to extend his current deal.
Bayern's sporting director Hasan Salihamidzic was unable to persuade the former Barcelona player to stay in Germany.
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Rummenigge told Germany's Sky Sport: "Actually, the conversations Hasan had with him were always very productive. And at some point it actually seemed that [a deal would be reached]. But then Thiago informed Hasan that he would like to do something new again, and so it is.
"We will have to accept that. We have a contract that is still valid for one year and if he agrees with any club and that club should pay a transfer of a certain amount.
"I don't want to make any public comment about the amount that we have in mind. Then we will deal with it, but we will not, as I have already said, hold a summer sale.
"I'm not angry at all, I like him. I think he's a great footballer, a fine footballer and he's a good character too. And one shouldn't forget either: At 29, he is at an age where, if he wants to do something new, he has to do it now because if you are in your 30s, it will be more difficult to make transfers."
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Rummenigge was circumspect on the impact of coronavirus on club finances, and expects transfer fees to drop during this window.
"I think I have a very good network with many colleagues - regardless of whether it is Real Madrid, Barcelona, many English clubs, France, Italy - and I can say very clearly: Everyone groans, everyone suffers. Everyone has reduced income," he explained.
"I can also say one thing: the bigger clubs have bigger revenue reductions than the smaller ones. Of course, that is in the nature of things, because of course the income is also a bit more questionable for all of us and we used to have higher income."