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“He is focused entirely on working every day to represent Texas and defend conservative principles in the Senate,” said Cruz spokesman Sean Rushton when asked about the rumors.

Not all senators who arrive in Washington with high political aspirations have taken Mr. Cruz’s fast-track approach.

When Hillary Rodham Clinton first won a Senate seat from New York in 2001, she earned the nickname “Senator Pothole” for her rigorous focus on local — not national — issues in the first years of her term.

Sen. Marco Rubio of Florida, who also is mentioned frequently as a potential 2016 Republican presidential candidate, was relatively low-key in his first year in the chamber in 2011, waiting almost six months to give his “maiden speech” — the last of that year’s 13 freshmen to do so.

But unlike Mr. Cruz, Mrs. Clinton, who went on to run for president in 2008 and serve as secretary of state for the past four years, enjoyed a huge built-in donor base thanks to her eight years as first lady before her 2000 Senate win. And while Mr. Rubio had been whispered as a possible presidential running mate in 2012, he never was considered a serious contender for the top of the ticket.

Still, Mr. Cruz must build up a significant legislative record if he is going to be considered a viable presidential candidate, analysts say.

“If you look at the Marco Rubio example, he has been careful not to jump too far out ahead of himself and to create a track record of success,” Mr. Bonjean said. “You just never know what the [future political] environment is going to be, so you want to start laying down a record right now.”

While Mr. Cruz joined the Senate less than three days after its historic pre-dawn vote to approve the “fiscal cliff” package, he has tried to retroactively establish a voting record, saying he would have defied his party’s leadership and rejected the measure, which passed the Senate on a 89-8 vote.

Among the Republican “no” votes were Mr. Rubio and Sen. Rand Paul of Kentucky, another conservative who has been rumored to be considering a 2016 presidential run.

By seeking as much attention as he has, Mr. Cruz always runs a risk of a verbal stumble.

“What he’s doing on the Sunday morning talk shows in many ways is getting beyond the echo chamber and sort of testing the right notes. But it is a double-edged sword because you could wind up stepping on your own toes and generating some bad press,” Mr. O'Connell said. “But, so far, he’s demonstrated himself to be quite capable.”