The South Korean defender has attracted early interest from Italy should his time in Germany fail to reignite, but Bayern are prepared to give him the opportunity to turn things around.
This season Jonathan Tah and Dayot Upamecano have established themselves as a preferred centre-back pairing, leaving Kim fighting to regain his place.
While Bayern are open to the idea of selling him, they plan to review the situation later in the campaign before taking any firm stance.
Sources believe it remains plausible that Kim could still convince the club to keep him - but at the very least a strong run of performances could spark competition for his signature in the market.
Inter Milan and AC Milan are understood to be early frontrunners for Kim, who retains an excellent reputation in Serie A following his successful spell at Napoli.
Juventus have also been mentioned, but talk of their interest is being treated with caution for now.
Other clubs are expected to monitor developments as well.
It is understood that, within the Premier League, Chelsea and Tottenham are both watching with interest, while Liverpool are also believed to be keeping an eye on the situation.
Kim joined Bayern for around €50 million, and it would likely take at least that figure for any club to prise him away. As things stand, a move next summer appears more realistic than a transfer in January.
For now, the 28-year-old remains fully focused on winning back his place in the Bayern side.
