His office has also refused to explicitly rule out the possibility, telling The Washington Post that he “remains fully focused on his role representing Texans in the U.S. Senate," but not definitively addressing a 2016 bid.
In a statement released by his office Wednesday, Cruz sought to downplay the story, saying it was a "continued source of amazement that the simple fact that I am working hard with like-minded senators to keep my promise is seen as newsworthy and cause for wild speculation."
"In my short tenure, my focus has been — and will remain — on two things: fighting for conservative principles in the Senate, and working to help elect strong conservatives to win a majority in the Senate in 2014," Cruz said.
But Cruz notably again stopped short of absolutely ruling out a presidential bid.
Still, some of the things that have endeared Cruz to grassroots conservatives may complicate his attempts to win his party's nomination. Cruz revealed at a Tea Party rally that fellow Republicans were "yelling at us at the top of their lungs" during Senate lunches following his move to filibuster the gun vote. Cruz went on to call his GOP colleagues "squishes" and suggest they "don't like to be held accountable."
Cruz will also likely face a "birther" controversy of his own; he was born in Canada to an American citizen. While Cruz's office and the Congressional Research Office have said that he qualifies as a "natural-born citizen," as required by the Constitution, Cruz could face legal challenges of his eligibility.
This post was updated at 1:47 p.m.