(CNN) New Hampshire Gov. Chris Sununu said there's an argument to be made that former President Donald Trump is "not the influence he thinks he is" -- or even the front-runner for the 2024 Republican presidential nomination.
"When the former president announced that he's running for president a week after the election, everyone went, 'OK,'" Sununu told CNN's Dana Bash. "He's announcing he's running for president at his most politically weak point. Right? He's doing it from a point of weakness, from a point of whatever his own agenda is, but it's kind of just a blip on the radar."
In the same interview, Sununu made it clear that he thought the future of the GOP was not Trump -- "He's done his time. He's done his service. We're moving on" -- and even floated that Florida Gov. Ron DeSantis could be a stronger presidential contender.
"There's an argument to be made that someone like DeSantis could beat (Trump) in a primary today," Sununu said.
Of course, Sununu himself has been the subject of 2024 speculation.
But he insisted to Bash, "I'm not thinking about it right now. I'm really not."
Saying he's "not trying to be coy," Sununu, who won a fourth two-year term as governor last month, noted: "My focus right now is New Hampshire. It really is because it has to be. And I want it to be."
That said, he's not closing the door on a potential run for president.
"I'm open to everything, of course. Look, I'm open to what comes in that door next. I'm open to any opportunity that might come down the road. Not for me personally but for the state or the community," Sununu said.
In Concord, the governor's office is a stone's throw from the secretary of state's office, where presidential candidates register to be on the ballot for New Hampshire's primary.
"So if you were to run for president, you would walk out that door, walk down this hallway and all you have to do is go right there?" Bash asked.
Sununu jokingly replied, "Probably just do it on my lunch break at some point. I won't tell anyone."
Whether or not he jumps in the race, Sununu thinks governors, in general, are suited to run for president, based on their executive leadership skills.
"I think a lot of us are looking at governors that may run, and I think as governors, we all want to see a governor run and be successful. Governors are executives. I always say the ... US Senate and Congress has its place, but they don't have executive leadership skills," he said.
As for Trump, Sununu doesn't doubt the former president could still win the Republican nomination, even though he sees him at a point of political weakness. He acknowledged that a crowded primary field could fragment the anti-Trump vote, allowing the former president to win the GOP nod with less than 35% of the vote.
Asked by Bash if there was a conversation taking place among Republican governors who are potential 2024 candidates about some of them staying out of the race to stop Trump from winning, Sununu said there wasn't. He also didn't think such a conversation should be happening "because it's up to the voters."
Yet the governor concedes it's something potential candidates will have to consider at some point.
"Just speaking to the folks I've talked to that might be elected officials, I think everyone understands that there's that obligation, responsibility. If it's not happening, it's not happening. You got to get out," Sununu said.
Still, he doesn't think Trump's candidacy will deter anyone from running.
"He's not at a point of political strength right now. I don't think it's going to be going up. I think, unfortunately for the former president, it's only probably a slow downward spiral from here. Could he still win the nomination? Absolutely. Let's not fool ourselves of that. But he's not going to keep anybody out of this race," Sununu said.
Unlike other Republicans, the New Hampshire governor has not shied away from criticizing Trump. Yet he still escapes Trump's famous counterpunch. Sununu addressed how he established a straightforward relationship with Trump and his administration from the beginning of his presidency.
"Early on, when he became president, we made it really clear, both with him we had a relationship and with the White House, that look ... if he does things that deserve credit, I'll be the first one to stand up and give him credit," Sununu said. "But when you have a tone or something that I disagree with or he says things or does things I disagree with, I'm going to say that too."
Politics runs in Sununu's family: He is the 82nd governor of New Hampshire, while his father, John, was the 75th. His older brother, John E. Sununu, represented the state as a Republican in the House and Senate from 1997 to 2009.
The state house halls he ran through as a kid are where he now leads the Granite State.
"It's the same place, everything's really the same. The portraits are maybe a little different, or they've upgraded the furniture, I suppose, a little bit. But no, it's really the same place," Sununu said.
In 1989, when Sununu started high school, his father became White House chief of staff to President George H. W. Bush.
"I remember walking into my freshman year, and kids are sitting kind of at their lockers, early in the morning, waiting for class to start. And they're reading The Washington Post," Sununu told Bash. "Given that my dad or federal policy was at the front of a lot of those discussions, it just made it very awkward. Very tough time."
It became especially tough when his father became embroiled in controversy, eventually resigning from the Bush White House in 1991.
"If anything, that's where I said, 'Look, I'm never doing this political thing,'" Sununu said.
But it was actually his mother, Nancy, who inspired him to get into politics.
"My mom was on the school board, I think, before my dad even got involved in politics. That was really my intro to politics. My mom would bring us over to the library, Kelly Library in Salem, New Hampshire, and I'd sit on the floor and color while the school board meeting was going on, in the bottom of the library," the governor said.
Sununu -- the seventh of eight children -- is raising three kids of his own with his wife, Valerie, in Newfields, New Hampshire.