Klose decision to come this week
The 32-year-old, who has scored 61 goals in 108 caps for Germany, is out of contract this summer and appeared to be on his way out of Bayern after only being offered a one-year deal.
Bayern's executive board chairman Karl-Heinz Rummenigge is now more hopeful that a "creative solution" can be reached with Klose, who scored only once in the Bundesliga last season having started just nine games.
Rummenigge told Die Welt newspaper: "I have talked with his advisers in the past week and [coach] Jupp Heynckes has spoken again with Miro. I expect that we will get an answer this week on whether he wants to stay."
Klose has been linked with Serie A champions AC Milan and Spanish side Valencia if new terms with Bayern cannot be reached.