Lis Smith, political strategist and former senior communications director for Buttigieg's 2020 presidential campaign, tells TMZ ... Leo offered a private coaching session with Pete ahead of a nationally televised debate, but Buttigieg's pop culture knowledge was lacking and he didn't recognize what he had in front of him.

We're told Leo extended the offer back in October 2019, during a swanky private fundraiser for Pete at a Los Angeles mansion, which was crawling with some of Hollywood's biggest stars ... including Paris Hilton, Ashton Kutcher, Katy Perry and Orlando Bloom.

It's funny ... Lis says Pete appreciated the Hollywood support, and the millions of dollars in donations that came pouring into his campaign, but he was never one to closely follow pop culture and barely recognized the celebs.

While we're told most celebs, like Paris, just took selfies with Pete and exchanged pleasantries, Leo went a step further ... giving Buttigieg the ultimate offer during a convo in one of the mansion's back rooms.

Leo's pitch went like this ... he would help Pete with his speaking style and coach him up on the best way to deliver his talking points at the upcoming debate.

Obviously, Leo's played all kinds of roles in Hollywood ... though he's not especially known for being spontaneous ... remember, Leo's never even hosted 'SNL,' but he's still talented as hell.

Lis says she was ecstatic when Pete told her about Leo's offer, but when she asked if he got DiCaprio's digits, Buttigieg said he didn't. Lis says she was beside herself, but chalked it up to Pete not fully grasping the significance of Leo's offer ... because he's not a big pop culture guy.

Ultimately, Pete handled the debate well without Leo's help ... with Lis calling it a turning point in Pete's campaign, sending him toward the top of the polls in Iowa and New Hampshire.

Lis spills more political tea in her upcoming book, "Any Given Tuesday: A Political Love Story" ... but this Buttigieg nugget isn't in the copy.