The top Republican tells WVLK radio in Kentucky, "Well you obviously can not do that".
The full Axios interview will air on Sunday.
Such a step would be regarded as an affront to the US Constitution, which was amended 150 years ago to include the words: "All persons born or naturalized in the United States and subject to the jurisdiction thereof, are citizens of the United States". By extending citizenship to those born in the USA, the amendment nullified an 1857 Supreme Court decision - Dred Scott v. Sandford, which had held that those descended from slaves could not be citizens.
Trump dispatched more than 5,000 active-duty military troops to the southern border on Monday as a slow-moving human caravan of around 3,500 Central American migrants, including nursing mothers and children, trekked towards the USA border at least 1,000 miles away.
By extending citizenship to those born in the US, the amendment nullified an 1857 Supreme Court decision (Dred Scott v. Sandford), which ruled that those descended from slaves could not be citizens. "A foreign national living in the United States is "subject to the jurisdiction thereof" because he is legally required to obey US law".
Changing an amendment in the Constitution would require the support of two-thirds of the US House of Representatives and the Senate as well as the backing of three-quarters of US state legislatures at a constitutional convention.
The law excludes the children of foreign diplomats - and in the 1860s, it was meant to give peace of mind to immigrants.
Birthright citizenship means that any child born in the USA automatically becomes a citizen.
Trump said he discussed the idea with his counsel who he said informed him a constitutional amendment is not needed to end birthright citizenship.
The president says he has the power to act on his own. "Guess what? You don't", Trump told Axios, which first broke the story, declaring he can do it by executive order.
"No matter how much he may want to appeal to the alt-right, President Trump can't alter the Constitution through executive order. It's ridiculous, it's ridiculous and it has to end".
Graham noted that birthright citizenship is taken advantage of by not only illegal immigrants from Central America but also a number of people from Russian Federation, the Middle East, and other areas, who visit the United States while pregnant just to get citizenship for their baby.
Is America actually the only country to have Jus Soli?
A person familiar with internal White House debate says the topic of birthright citizenship has come up various times over at least the a year ago.