During an eyebrow-raising visit to New Hampshire on Friday, Senator Joe Manchin III of West Virginia name-checked friends who are elected officials in the Granite State and complimented the discerning nature of its voters.

He paid homage to the state’s first-in-the-nation primary tradition and swiped at President Biden’s decision to undercut New Hampshire’s power in this year’s Democratic contest.

And when pressed on his own ambitions, the conservative Democratic senator offered a message that would-be candidates have often deployed as they flirt with this historically influential early-voting state: He declined to rule anything out.

“How would you feel if a bunch of Democrats in New Hampshire wrote in ‘Joe’ — not Biden — but wrote in ‘Joe Manchin’?” an attendee asked as Mr. Manchin kicked off a “listening tour” at Politics and Eggs, an event series at the New Hampshire Institute of Politics that has long hosted presidential candidates and potential contenders.