|Fawkes the phoenix|
J.K. Rowling had this to say on the matter: The idea that anybody believes this is strangely upsetting to me.
Why would Harry Potter fans believe this anyway? One reason is because Fawkes seems to always appear during a time of need. Another reason is because Dumbledore seems to understand an awful lot about the Dark Lord's relationship with his Horcrux, Nagini. For example, Dumbledore liked to keep Fawkes close just like Voldemort kept Nagini close. Similarly to Lord Voldemdort, Dumbledore also seemed to have an unusual amount of control over Fawkes.
These are reasons that fans might believe the theory. However, there are also reasons that fans should not believe this theory. For example, one reason that this theory isn't a good one is because Dumbledore did not approve of Dark Magic. However, some fans argue that just because he disapproved of Dark Magic when we met him doesn't mean Dumbledore was always that way. Secondly, Dumbledore hasn't knowingly, intentionally killed someone which is crucial to making a Horcrux. Lastly, another reason you shouldn't believe this theory is because J.K. Rowling flat out denied it.
My opinion? Dumbledore probably didn't make any Horcruxes. I get the feeling that other fans just don't want to accept that Dumbledore is truly dead.