Ted Cruz has introduced a bill to limit US senators to two terms in office, thereby removing from Washington what he calls “permanently entrenched politicians … totally unaccountable to the American people”.

On Sunday, however, he said he saw no problem with running for a third term himself.

“I’ve never said I’m going to unilaterally comply,” the Texas senator said.

Cruz was speaking to CBS’s Face the Nation.

Elected to the US Senate in 2012, Cruz emerged as a face of the Republican hard right through stunts including reading Dr Seuss and impersonating Darth Vader during a marathon floor speech and prompting a government shutdown.

Such behaviour did not make him popular in Congress. Al Franken, then a Democratic senator, once said, “I like Ted Cruz more than most of my other colleagues like Ted Cruz. And I hate Ted Cruz.”

Nonetheless, Cruz challenged strongly for the Republican presidential nomination in 2016, finishing the primary second to Donald Trump.

After a brief spell as a rightwing alternative to Trump, Cruz won a second term in 2018 despite a strong challenge from the Democrat Beto O’Rourke.

Cruz’s name now features, if not strongly, in polling regarding the notional field for the Republican presidential nomination in 2024 – when he will be up for Senate re-election.

Congressional term limits are a popular policy offering on the American right.

Introducing his bill to achieve a constitutional amendment, an effort mounted with Ralph Norman, a House Republican from South Carolina, Cruz said term limits were “critical to fixing what’s wrong with Washington DC”.

Bemoaning “a government run by a small group of special interests and lifelong, permanently entrenched politicians … totally unaccountable to the American people”, he said: “Terms limits brings about accountability that is long overdue”.

On Sunday, his CBS host, Margaret Brennan, said: “You introduced a bill to limit terms to two six-year terms in office for senators. Why aren’t you holding yourself to that standard? You said you’re running for a third term.”

Cruz said: “Well, listen, I’m a passionate defender of term limits. I think that Congress would work much better if every senator were limited to two terms, if every House member were limited to three terms. I’ve introduced a constitutional amendment to put that into the constitution.”

Brennan said: “But you’re still running.”

Cruz said: “And if and when [the term limits amendment] passes I will happily, happily comply. I’ve never said I’m going to unilaterally comply. I’ll tell you what, when the socialists and when the swamp …”

Brennan interrupted, asking: “Are you running for president?”

Cruz carried on, saying “… are ready to leave Washington, I will be more than happy to comply by the same rules that apply for every one. But until then, I’m going to keep fighting for 30 million Texans because they’ve asked me to do” so.

Brennan said: “I think you’ve heard me ask if you’re running for president.”

Cruz said: “I’m running for re-election to the Senate. There’s a reason I’m in Texas today. I’m not in Iowa, I’m in Texas, and I’m fighting for 30 million Texans.”

In 2018, 4.26 million Texans voted to send Cruz back to the Senate. More than 4 million voted to restrict him to one term.