The German club are prepared to sell the defender for around €40 million as they look to generate funds before making further additions to their squad.
Kim has not been a regular part of Vincent Kompany's plans and Bayern are understood to believe that this is the right moment to move him on.
A departure would also remove a significant salary from the club's wage bill, with Kim reportedly earning close to €8m per year.
Interest has already emerged from England, with Tottenham and Villa having made enquiries about the centre-back. AC Milan have also asked about his situation, although the Italian club have not submitted an offer.
Bayern's willingness to sell Kim is significant because the club are also monitoring the market for potential defensive changes.
Their approach is understood to be based on raising money through player sales before committing to major investment, making Kim one of the players whose departure could help fund their next moves.
For Tottenham and Aston Villa, the availability of an experienced international defender at a clearly stated asking price provides an opportunity to assess whether a deal can be reached.
Kim joined Bayern after establishing himself as one of the leading defenders in European football and remains under contract with the Bundesliga club.
His current position, however, has changed under Kompany, with limited involvement increasing the possibility of a move.
Bayern Munich's €40m valuation gives interested clubs a clear indication of what it would take to secure the defender, while Milan's interest means the Premier League clubs are not the only teams monitoring developments.
Tottenham and Villa have both now been linked with the South Korean, although neither has made a formal offer at this stage.
Bayern's decision to make Kim available could lead to more movement in the coming weeks as the German club attempt to complete a sale before strengthening elsewhere.