Are we talking about Mashiach ben Yosef or Mashiach ben David? I heard from a rav once that Mashiach ben Yosef doesn’t need necessarily need to be a direct father to son descendant from Yosef haTzadik (i.e. he could be of matralinial descent) but I don’t know a source on the inside for this.
As far as Mashiach ben David we know he needs to be a direct patralinial descendant of David HaMelech so let’s look at who has a mesorah for that. Rav Daniel Glattstein said a few days ago in a TorahAnytime shiur that Rashi was a direct descendant (and apparently he’s a direct descendant of Rashi so I guess he’s a candidate!) – so anyone who can trace father to son back to Rashi.
The Baal Shem Tov, Rebbe Nachman m’Breslov, and the Maggid m’Mizritch also were direct descendants so somebody who could trace lineage back to them would be in the running as well. Rav Yosef Shlomo Dayan came from a large Syrian family in which they apparently had documents recording their family back to David Hamelech as well. As he never married he didn’t have any children though I believe he had a lot of relatives.
These are the ones that come off the top of my head – is anyone else familiar with a lineage like these?