Catherine Frazier, a spokeswoman for the junior senator from Texas, said lawyers are preparing the necessary paperwork.
The 43-year-old Cruz "has been fully focused on fighting for Texans' values and interests in the Senate for the last year," Frazier said via email. "He looks forward to the process being completed soon."
Cruz hasn't even been in office for a year, but already helped spark the budget fight that led to a partial government shutdown in October, and is being mentioned as a possible 2016 Republican presidential contender.
Frazier's response comes after Canadian immigration attorney Richard Kurland suggested Friday that the process was relatively simple and quick. Kurland wondered what was taking Cruz so long.
Amid questions this summer about his eligibility for the White House, Cruz released his birth certificate in August to the Dallas Morning News and pledged to renounce his Canadian citizenship. Cruz said then that his mother had been told he would have to take affirmative action to claim Canadian citizenship - and the fact he automatically received it at birth was news to him.
"If he's attempting to bring our system into disrepute by suggesting it's lengthy and complex, it's just not true," said Kurland, who's based in Vancouver. "Revocation is one of the fastest processes in our system."
The citizenship issue is a thorny one for Cruz, since some conservatives accused President Barack Obama of being born in Kenya and thus not eligible to be U.S. president. Obama is an American citizen; his father was Kenyan, his mother American.
Cruz was born in Calgary, Alberta, in 1970, while his parents were working in the Canadian oil business. His mother, Eleanor, was born in Delaware, while his father, Rafael, is a Cuban who didn't become a U.S. citizen until 2005.
The U.S. Constitution says only a "natural born Citizen" may be president. Legal scholars, though, generally agree the description covers foreign-born children of U.S. parents. Canada, like the United States, gives automatic citizenship to anyone born on its soil.
Previous foreign-born Americans - notably Republicans John McCain and George Romney - have run for president with some mention but no serious challenges of their eligibility.
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