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Mr. Madden said grass-roots conservatives “respond to Cruz because they see him as taking the fight directly to Obama with an almost unrivaled passion.”

“They like to see that kind of fight in a candidate,” he said.

It’s that same take-no-prisoners approach, though, that has Democrats licking their chops over the idea of a Cruz candidacy.

“If he gets into the race, Republicans everywhere will regret this day for a long time,” said Jim Manley, a Democratic strategist. “His brand of politics is toxic for the party.”

Political observers say it will become more clear that Mr. Cruz is running for president once he begins to smile more on the stump, changes his hairstyle and “tones down the fire-and-brimstone rhetoric.”

Should Mr. Cruz enter the fray, he would be following in the footsteps of Mr. Obama, who ran during his first term in the Senate — a move that some say helped shield the Democrat from having to defend a long voting record.

That same strategy could help Mr. Cruz, who likely would be competing with the likes of former Sen. Rick Santorum of Pennsylvania, whose long voting record in Congress came back to haunt him in the 2012 GOP primary.

Heading into his Friday visit to New Hampshire, Mr. Cruz remains a relative unknown to many.

The latest WMUR Granite State Poll, conducted by the University of New Hampshire Survey Center, shows that 4 percent of likely Republican primary voters would support him — putting him squarely behind New Jersey Gov. Chris Christie’s 21 percent, Sen. Rand Paul’s 16 percent and former Florida Gov. Jeb Bush’s 10 percent.

The same poll, though, shows people are becoming more familiar with Mr. Cruz. In July, 48 percent of likely Republican primary voters did not know of Mr. Cruz — down from 62 percent in February. His favorability also jumped over that time, from 18 percent to 29 percent.

Mr. Cruz has been asked to speak in New Hampshire this fall at an annual fundraising dinner hosted by Cornerstone Action, an influential conservative organization that promotes “traditional values, limited government, and free markets.”