Their bid is a loan deal with an option to buy him next summer.
Bayern are in the market for a new keeper after news broke that Manuel Neuer would not be fit for the start of the new season.
The German international suffered a leg break during a skiing tour that ruled him out for the second half of last season.
Neuer returned to light training in pre-season but discovered he is not ready. The Bayern stalwart could be out for months.
Die Rekordmeister recently let Yann Sommer go after Inter Milan triggered his release clause, leaving the German club light between the sticks.
They have now turned to Arrizabalaga, reportedly on the orders of manager Thomas Tuchel.
Tuchel managed Arrizabalaga at Chelsea and knows his quality, even though he was mostly the second choice to Edouard Mendy.
Chelsea have been trying to sell Arrizabalaga for some time now but have struggled to find suitors ready to take on his £190,000 per week wages.
They appear to have found a willing buyer, but it is unclear whether they will accept Bayern’s offer.
The Blues have signed Robert Sanchez and are already reportedly eyeing a move for Valencia’s Giorgi Mamardashvili and Borussia Dortmund’s Gregor Kobel.
They wanted to sign him in January 2018, but Zinedine Zidane was against the idea, and Chelsea signed him when Courtois went to Real Madrid that summer.
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