Justin Bieber -- Meet My Clean-Cut Mentor ... Floyd Mayweather!

Justin Bieber's appearance with Floyd Mayweather last night was months in the making -- and sources close to The Biebs tell TMZ ... Justin feels his new "mentor" is really just "misunderstood."

According to our sources, Mayweather asked Bieber four months ago to join him in Las Vegas for the Miguel Cotto fight ... and Bieber, a huge boxing fan, jumped at the opportunity.

We're told Floyd and Justin became friends after Floyd saw Justin's movie, "Never Say Never," and reached out to his camp. According to our sources, Floyd decided he wanted to mentor Justin and the two developed a relationship.

Our sources say Mayweather tried to impart his work ethic on Justin, basically telling him he could develop the same level of "greatness" Floyd achieved ... if he worked as hard as Floyd did.

We're told the two hit it off and Justin tells people he thinks Floyd is just misunderstood ... and that he is a really a "hard-working and clean-cut guy."

Several judges would disagree, but whatever ...