Justin Bieber is gonna EAT LIGHTNING AND CRAP THUNDER ... 'cause he's now training to box with the greatest fighter alive -- Floyd Mayweather, Jr.

Bieber and Floyd threw on the gloves at the legendary Ten Goose boxing gym in Van Nuys, CA -- where some pretty big fighters have trained ... guys like Sugar Shane Mosely, Andre Berto and David Tua.

Sporting a pair of L.A. Clippers shorts, Bieber worked on his jab ... and moving around ring -- making sure to document every single second on his Shots page.

Bieber also hit the speed bag -- but curiously, there was no video of that ... just a pic.

So, will Bieber ever get in the ring and actually fight someone?

We can only hope.

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