The pugnacious governor of Maine, who has weathered storm after storm since rising to office on a Tea Party wave in 2010, hinted Tuesday that he may step down after unleashing a barrage of vulgar, threatening, and racially charged remarks.

In a radio interview, Paul R. LePage apologized for threatening a state legislator last week. He also openly questioned whether he was up to the job of leading the state, saying, “I think some things I’ve been asked to do are beyond my ability.’’

But hours later, the volatile governor seemed to backtrack.

“Regarding rumors of resignation, to paraphrase Mark Twain: ‘The reports of my political demise are greatly exaggerated,’ ’’ he wrote on Twitter.

The uncertainty exasperated leaders of both parties, who said LePage crossed a line when he mused publicly about shooting a Democratic state lawmaker and blamed Maine’s opioid crisis on blacks and Hispanics coming from out of state.

“This is a man on a mission to invoke fear and violence,’’ said Justin Alfond, the Democratic leader of the state Senate. “This is man who is unfit, and his actions and his words are disturbing and offensive.’’

LePage’s constituents were openly divided about the best path forward, with many saying he should resign.

“I don’t feel that he’s helping Maine in any way, to be honest,’’ said Jamie Ploss, a 37-year-old Bradford resident, who was in downtown Bangor.

But car mechanic Al Scott, 48, exclaimed, “Absolutely not!’’ when asked whether LePage should quit.

“I think he’s the best governor we’ve ever had,’’ Scott said as he worked on a car hoisted on a lift. “He speaks his mind. Any Republican or independent who speaks their mind, they’re branded a racist.’’

If LePage were to resign, Michael D. Thibodeau, the Republican Senate president, would serve as governor until the end of LePage’s term in January 2019.

Thibodeau and other Republican leaders met with LePage on Monday night at Blaine House, the governor’s mansion in Augusta.

“As a result of that meeting, the governor said he would be talking to his family and closest friends about corrective action that was discussed in the meeting,’’ said James Cyr, a Thibodeau spokesman. “The ball is now in his court.’’

LePage is no stranger to political firestorms, having survived lawsuits, an impeachment attempt, and a long string of comments he made that have been criticized as vulgar and racially insensitive.

But his problems seemed to escalate over the last several days, beginning last Wednesday, when he said he keeps a three-ringed binder of photos from drug busts, and most ‘‘are black and Hispanic people from Waterbury, Conn.; the Bronx; and Brooklyn.’’

The following day, LePage left an obscenity-laced voicemail for state Representative Drew Gattine, a Democrat he accused of calling him a racist, a charge Gattine denies.

“I’m after you,’’ LePage said on the voicemail, before telling reporters he wished it were 1825 so he could challenge Gattine to a duel and point a gun between his eyes.

Democrats condemned the threat and urged Republicans to get LePage “professional help.’’

“Anyone who heard that voicemail heard a man who clearly is unhinged, enraged, and not in control of his emotions, his judgment, or thinking,’’ Alfond said.

Said House Speaker Mark Eves, a Democrat: “We’re not professionals, but we do feel he needs professional help, whether it’s anger management or if there’s something more going on, like a substance abuse problem.’’

State Senator Amy Volk, a Republican, also questioned on Facebook whether LePage was suffering from “substance abuse, mental illness or just ignorance.’’

But she said she did not believe he should resign.

“I just feel like we really need to see a change in communication, and the outbursts are really inappropriate,’’ she said Tuesday. “They’re hindering our ability to get things done.’’

LePage’s staff did not immediately respond to the suggestions of substance abuse or illness.

Tuesday night, House Republican leader Ken Fredette defended LePage’s record at a press conference but said the governor needs to offer a “sincere apology’’ to the people of Maine.

“He’s abrasive and rough on the edges and no one agrees with that,’’ Fredette said. “I get so mad when he does that because I believe it takes away from the good work that we are doing.’’

In his radio interview Tuesday, LePage, 67, said he didn’t know if he would complete his term but was “looking at all options.’’

“I’m not going to say that I’m not going to finish it. I’m not saying that I am going to finish it,’’ he said.

He alluded to the criticism he has faced from fellow Republicans.

“It’s really one thing to have one party behind you,’’ he said. “It’s another thing not to have any party behind you.’’

LePage expressed regret for threatening Gattine, and said he wants to meet with him in person.

“There’s no excuse; it’s unacceptable,’’ he said. “I just apologize to the Maine people, to Gattine’s family, and most of all to my family.’’

He said he was so angry when he was called a racist, he couldn’t breathe.

“It’s like calling a black man the ‘N’ word or a woman the ‘C’ word,’’ he said. “It just absolutely knocked me off my feet.’’

But LePage did not apologize for blaming Maine’s opioid problem on blacks and Hispanics coming from out of state.

“I got all of my information in my book from the press. It’s that simple,’’ he said. “Every drug arrest, we get the story and the people. And when it comes to meth labs, they’re essentially all Maine white people. When it comes to the heroin epidemic, it’s just the opposite.’’

Christina Prignano and Brian MacQuarrie of the Globe staff contributed to this report. Michael Levenson can be reached at Michael.Levenson @globe.com. Follow him on Twitter @mlevenson.