And some in the conservative movement are already talking him up as a potential 2016 presidential candidate.
"It is an undercurrent and growing," said Texas GOP consultant Bill Miller. "Most observers believe he is moving very fast toward a much higher visibility. The presidential buzz has begun in earnest."
Jeff Judson, a major tea party figure in the Lone Star State and a Cruz booster, said the movement is totally organic and not emanating from Cruz or his staff.
"Yes, there is a buzz about Cruz running for president, but a different kind of buzz than with other prior candidates," Judson said. "He has serious gravitas -- more than all the 2012 Republican presidential candidates combined."
The question from here is two-pronged: (1) Is the Canadian-born Cruz actually eligible to run for president? And (2) What would his campaign look like?
1. Cruz's eligibility
Cruz, like a couple of presidential candidates before him, faces a potential hurdle to running for president in that it's not 100 percent clear that he's a "natural-born citizen," as the Constitution requires presidents to be.
Cruz's mother was a U.S. citizen when he was born, and current U.S. law extends citizenship to anyone born to a U.S. citizen, regardless of where the birth takes place. The question is whether citizenship is the same thing as being a "natural-born citizen."
Legal scholars generally agree that Cruz meets that requirement, and Cruz's office agrees. But it also remains somewhat untested in the courts.
While no president-elect has formally tested the "natural-born citizen" requirement, several have run for president with that question hanging over their candidacies.
Democrats in 1967 suggested that George Romney would not be eligible to serve as president, because he was born to U.S. citizens in Mexico. But a New York Law Journal piece at the time argued forcefully that he would be, and that seemed to put the issue to rest. (Romney's primary campaign wound up imploding.)
Sen. John McCain (R-Ariz.), the GOP's nominee in 2008, was born in the Panama Canal Zone to U.S. citizens. After he secured the party's nomination, the Senate in 2008 passed a resolution stating that McCain was indeed a natural-born citizen.
In fact, this debate dates back to President Chester A. Arthur and the original so-called "birther" controversy. While Arthur is listed as being born in Vermont, some argued that he was born in Canada and thus ineligible to be president.
The nonpartisan Congressional Research Service has even weighed in on the issue, writing in November 2011 that people born to U.S. citizens in foreign countries "most likely" qualify as natural-born citizens.
"The weight of more recent federal cases, as well as the majority of scholarship on the subject, also indicates that the term 'natural born citizen' would most likely include, as well as native born citizens, those born abroad to U.S. citizen-parents, at least one of whom had previously resided in the United States, or those born abroad to one U.S. citizen parent who, prior to the birth, had met the requirements of federal law for physical presence in the country," wrote Jack Maskell.
2. Where Cruz fits
So let's assume Cruz can become president. The question is will he?
Cruz's office won't rule out the possibility. Spokeswoman Catherine Frazier said Cruz "remains fully focused on his role representing Texans in the U.S. Senate" but didn't directly address whether he would rule out a 2016 run.
Cruz has clearly been angling for a high profile.
The senator built a national following even before he joined the Senate, having been a cause celebre for the conservative and tea party movements in his 2012 runoff campaign against Texas Lt. Gov. David Dewhurst -- the well-funded establishment favorite.
More recently, he was one of the first senators to join the 13-hour filibuster of Sen. Rand Paul (R-Ky.) against President Obama's drone program, and he made waves during a sharp exchange with Sen. Dianne Feinstein (D-Calif.) while the Senate Judiciary Committee was debating gun legislation last week.
His outspoken conservatism earned him the keynote speaking slot at last week's Conservative Political Action Conference, and his speech about how conservatives were winning the debates in Washington was a hit with attendees.
Come 2015, Cruz will have been in the Senate just as long as President Obama was when he announced his presidential campaign. So it would hardly be unprecedented for Cruz to run.
His eligibility and newcomer status aside, though, there is plenty to suggest that Cruz won't run in 2016. The two main things that make Cruz attractive as a candidate are his strongly conservative record and his potential to appeal to Hispanics as a Cuban-American Republican. But the potential 2016 field already has more-established senators who check those boxes.
Rand Paul, who is very much on the same page with Cruz ideologically, is clearly ramping up for a potential presidential bid. And Sen. Marco Rubio (R-Fla.), another Cuban-American GOP senator, is seen as the early front-runner for the GOP nomination.
If both of those senators wind up running in 2016, Cruz may have a hard time carving out his niche in the presidential field. And indeed, it seems unlikely that Cruz would run against his close ally Paul.
At the same time, Cruz is also the guy who could check both of those boxes at once. And if either Paul or Rubio stay out of the race -- or if both decide not to run -- it wouldn't be surprising to see a formal push for Cruz to get in. That may not come from the GOP establishment, which is wary of the outspoken Cruz, but it would be a push nonetheless.
Still, some think it's too early for the new senator.
"He's more viable than folks might think because there was such a national effort to elect him," said one Texas GOP consultant, granted anonymity to offer a candid assessment. "But even so, no one who is serious is talking about a Cruz presidential in '16."
A more likely option for Cruz is to become the new Jim DeMint of the Senate. DeMint, who had no presidential ambitions, carved out a niche by pushing for more conservative Republican candidates, all the while antagonizing the GOP establishment.
With DeMint now heading the conservative Heritage Foundation, there is a void in the Senate that Cruz seems like a natural fit to fill.
There's still a lot of time between now and 2016, though, and we would expect there to be plenty of speculation about Cruz running for president between now and then.