The Denmark international has just over four months remaining on his Blues deal, and it appears likely that he could leave on a free transfer this summer.
Christensen can hold discussions with foreign clubs, and it is now reported that both Barcelona and Bayern have entered talks over a Bosman switch.
The 25-year-old is eyeing a foreign challenge over joining another Premier League club out of respect for the Blues where he graduated from.
Christensen was close to penning a new deal with the Blues last summer, but he had a sudden change in stance amid his agent demands and contract length.
The centre-back has since changed his agency, and it was recently claimed that Barcelona could be the front-runners to land his signature this summer.
However, Bayern have now entered the pursuit. The Munich giants could have the edge as they could assure him with a regular starting berth next term.
They are set to lose Niklas Sule on a free transfer to Borussia Dortmund this summer, and Christensen could be an ideal replacement for the German.
Christensen spent two seasons on loan at Borussia Monchengladbach between 2015 and 2017, where he made a big name for himself in the Bundesliga.
His experience could be the driving factor behind Bayern’s pursuit. They could also be planning for low-cost signings amid a possible change in transfer policy.
With manager Julian Nagelsmann recently experimenting with a back three, he appears ideally suited for the club as he is accustomed to the tactical system.
He could be competing with Dayot Upamecano, Lucas Hernandez and Benjamin Pavard for a starting role.
It could be a different story with Barcelona, with manager Xavi Hernandez preferring to play with a back four since his appointment in November.
Unlike Bayern, they are loaded with central defensive options in Gerard Pique, Eric Garcia, Ronald Araujo, Clement Lenglet, Samuel Umtiti and Oscar Mingueza.
A couple of them could be shown the exit door during the summer, but Christensen may still struggle to nail down a starting spot with the Catalan club.
Only time will tell where he will be playing next term. Bayern seem a more suitable destination over Barcelona, but it ultimately depends on the player.