In the meeting, the Republican also cracked wise about Baptists, saying "they're the ones who drink and don't admit it," a variation of a joke he sometimes tells in public.
Graham's folksy sense of humor and his shoot-from-the-hip style are part of his political appeal, and well-known to journalists and his colleagues in the Senate. But the behind-closed-doors remarks are a departure from the G-rated jokes he tells in public, and another reminder for politicians in the iPhone era that they are rarely far from a microphone, no matter how private the setting.
The audio snippets were provided to CNN on Wednesday by two separate South Carolina Democrats who received the recordings from a person using an anonymous Gmail address. Graham confirmed the recordings in an interview Wednesday with CNN.
The person didn't provide details on his background but told the Democrats that Graham was speaking to an "all-male club" in Charleston earlier this month, the Democrats said. Both of the Democrats wished to remain anonymous.
After CNN contacted his campaign, Graham said in the interview that he made the comments several weeks ago at a meeting of the Hibernian Society of Charleston, the Irish-Catholic charity organization that sometimes hosts big-name politicians for irreverent and frequently bawdy speeches.
Graham said the Hibernian Society encourages speakers "to be earthy, to make fun of yourself, to make fun of them. Then you say something serious and sit down. And if you talk over 20 minutes, they throw something at you."
In the recordings, Graham, who is favored in his race for re-election next Tuesday against Democrat Brad Hutto, appears to joke about the possible 2016 presidential bid he recently floated in an interview with the Weekly Standard.
"I'm trying to help you with your tax status," Graham says in the recording. "I'm sorry the government's so f---ed up. If I get to be president, white men in male-only clubs are going to do great in my presidency." The crowd is then heard laughing.
In another clip, Graham asks, "We got any Presbyterians here? We got any Baptists? They're the ones that drink and don't admit it." Again, the crowd chuckles.
Contacted about the recording, Graham's campaign arranged for the senator to speak about the remarks by phone from South Carolina.
The Hibernian Society hosts an annual St. Patrick's Day banquet that, in the past, has been headlined by top political figures like Vice President Joe Biden and former Vice President Dick Cheney.
"I have never been to the Gridiron Dinner, but it's a little like that," Graham said.
His October visit to the society was a separate event, he said, but the spirit was the same.
"I hope this doesn't reflect poorly on the club," he said. "They are great guys."
Graham said he was making fun of the society's all-male membership, joking that they were the last such organization in existence after Augusta National Golf Club admitted women.
As for the Baptist riff, Graham said he regularly teases South Carolina audiences about their various religious denominations, before making a larger point about religious liberty in the United States and warning about the threat of radical Islam overseas.