Sid Miller, currently the Agriculture Commissioner for Texas, has been a vocal backer of Mr Trump’s policies on immigration and his desire to construct a wall on the US’ southern border.
In recent days Mr Trump has moved to enact some of his more extreme campaign promises.
Speaking to Nick Robinson on BBC Radio 4’s Today programme, the 61-year-old was asked: “You once said ‘you stayed awake at night worrying about the United States would become a Muslim country. Is that right?
Mr Miller replied: “I did. The question was ‘what keeps you up late at night?’”
He continued: “The thing that I worry about when I hold my grandkids on my knee what the future holds for them.
“Will we no longer be a republic and become a socialist country?
Ted Miller has been a vocal supporter of some of Mr Trump's policies and campaign pledges
“Will we no longer be a Christian country, will we be a Muslim country?
“So the future of my grandchildren is what sometimes keeps me up.”
Currently, the Muslim population of the United States is around one per cent and Robinson asked if the Texan’s fears were anything more than “paranoia”.
Mr Miller then pointed to the fact the Muslim population of both the UK and Europe had risen in recent years.
However, Robinson rebuffed his point by stating that while that was true, neither the UK or Europe had become “Muslim countries”.
“But the population is rapidly expanding,” Mr Miller replied.
Mr Trump has moved quickly to enact certain campaign pledges
Will we no longer be a Christian country, will we be a Muslim country?
“In a way you think is dangerous?” Robinson asked.
“I do,” Mr Miller said. “We need to at least keep the dangerous ones out.
“I have no problem with a law-abiding citizen who moves here to better their family or their lifestyle. That is part of the American dream.”
Donald Trump is set on Thursday to suspend the US refugee programme for four months.
His administration is also poised to put a block on visas for travellers from seven countries which have high Muslim populations.
Visa applications from countries deemed a terrorist threat – Iraq, Syria, Iran, Sudan, Libya, Somalia and Yemen – will be halted for 30 days.
The Mexico Wall: Trump gives the go ahead
Thu, January 26, 2017
The structures, fences and walls that mark the border between the United States and Mexico as President Donald Trump reiterates his promise to build a wall along the U.S. border with Mexico.
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Combo with images of the border fence between Mexico and the U.S. taken in the states of Sonora and Baja California, Mexico, and in the states of Arizona and California, U.S., between July 2, 2016 and January 25, 2017.
A draft executive order, published in the Washington Post, said refugees from war-torn Syria will be indefinitely banned.
Mr Trump told ABC News late Wednesday that his plan to limit the entry of people from Muslim countries was necessary because the world is "a total mess."
"No it's not the Muslim ban, but it's countries that have tremendous terror," Trump said. "And it's countries that people are going to come in and cause us tremendous problems."