President Trump claimed last week that "some" former presidents privately confided to him that they support his mission to build a border wall. As of Monday, every living president has said otherwise.

Former President Jimmy Carter issued a statement denying that he and Mr. Trump had any such conversation. He now joins former Presidents Bill Clinton, George W. Bush and Barack Obama in either refuting Trump's claim or denouncing his demands for the wall.

"I have not discussed the border wall with President Trump, and do not support him on the issue," Mr. Carter said in a statement issued by the Carter Center.

Clarification from former U.S. President Jimmy Carter: "I have not discussed the border wall with President Trump, and do not support him on the issue.” -- Jimmy Carter — The Carter Center (@CarterCenter) January 7, 2019

A spokesman for Mr. Bush told Politico that the 43rd president never talked to Mr. Trump about a border wall. A spokesman for Mr. Clinton also denied it and said the two haven't spoken since Mr. Trump's inauguration.

Get Breaking News Delivered to Your Inbox

Mr. Obama is the only living president who has not explicitly denied having this conversation, and his office did not return a request for comment from CBS News. But Mr. Obama has repeatedly spoken out against Trump administration immigration policies and made clear since the 2016 campaign that he does not support a proposed wall at the U.S.-Mexico border. Politico also pointed out that Mr. Obama and his successor have not had any extensive conversation since the 2017 inauguration.

Acting White House Chief of Staff Mick Mulvaney told CNN he had "no idea" which past presidents told Trump they supported the wall. And former Vice President Joe Biden said he "can't think of a single" past president who wanted a border wall.

President Donald Trump said other presidents have told him "they wished they had built a wall." Former Vice President Joe Biden can't think of a single one.https://t.co/q1hTBws4eI pic.twitter.com/H3yUp59Bpm — Circa (@Circa) January 5, 2019

Past presidents have supported various measures to curb illegal immigration and secure the U.S.-Mexico border, including with strong fencing in some areas, though none of those policies resembled Trump's proposal for a multibillion-dollar steel wall covering most of the border. In 2006, then-Senator Obama voted for and President George W. Bush signed the Secure Fence Act, which approved about 700 miles of fencing along parts of the border. Trump has dismissed that fencing as a "nothing wall."

The government shutdown, now in its third week, started when Mr. Trump refused to sign congressional spending measures without $5 billion for the wall. Trump said in a news conference last Friday, "This should have been done by all of the presidents that preceded me, and they all know it. Some of them have told me that we should have done it."