President Trump announced a trade "understanding" with Mexico Monday that could lead to an overhaul of the North American Free Trade Agreement, and also said he wants to get rid of the name "NAFTA."

"They used to call it NAFTA," Mr. Trump said in the Oval Office. "We're going to call it the United States-Mexico Trade Agreement. We'll get rid of the name NAFTA. It has a bad connotation because the United States was treated very very badly for NAFTA."

It's unclear yet if Canada will be a part of any revised agreement — Mr. Trump said "we'll see" if Canada can be a part of the deal with Mexico, or if the U.S. and Canada will need a separate deal. Mr. Trump threatened tariffs on cars if Canada doesn't come to an agreement.

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"We're starting negotiations with Canada pretty much immediately," Mr. Trump said, adding he will be speaking with Canadian Prime Minister Justin Trudeau "in a little while."

The president would need Congress' approval for any new trade agreement.

Mr. Trump, surrounded by reporters, made the announcement by inviting Mexican President Enrique Peña Nieto on the phone, although the phone call placed on speaker was plagued with technical difficulties.

"I thought we would congratulate each other before it got out, and I know we will have a formal news conference in the not-too-distant future," Mr. Trump told Peña Nieto.

Mexico is holding a news conference Monday afternoon on what it describes as trade agreement principles reached on "NAFTA."

The commander-in-chief ignored reporters' questions Monday, including ones about the legacy of Sen. John McCain following his death Saturday. Mr. Trump reportedly nixed a statement praising McCain, whom he sparred with, and the flags at the White House were at full staff Monday morning when flags at other Washington landmarks were still at half staff.

Shortly before the last-minute call in the Oval Office Monday, Mr. Trump foreshadowed his announcement.

"A big deal looking good with Mexico!" Mr. Trump tweeted, making no mention of Canada.

A big deal looking good with Mexico! — Donald J. Trump (@realDonaldTrump) August 27, 2018

Nieto tweeted just before Mr. Trump's announcement that he spoke to Trudeau about the status of NAFTA negotiations and progress the U.S. and Mexico have made, and expressed the importance of Canada's involvement in the negotiations.

Hablé con el Primer Ministro de Canadá, @JustinTrudeau, sobre el estado de las negociaciones del TLCAN y el avance entre México y EUA. Le expresé la importancia de su reincorporación al proceso, con la finalidad de concluir una negociación trilateral esta misma semana. — Enrique Peña Nieto (@EPN) August 27, 2018

Canada pointed out that its signature is required on any new deal.

"Canada is encouraged by the continued optimism shown by our negotiating partners," said Adam Austen, spokesperson for Chrystia Freeland, Canada's minister of foreign affairs. "Progress between Mexico and the United States is a necessary requirement for any renewed NAFTA agreement. We are in regular contact with our negotiating partners, and we will continue to work toward a modernized NAFTA. We will only sign a new NAFTA that is good for Canada and good for the middle class. Canada's signature is required."

In May, Canada rejected a proposal Mr. Trump wanted to add to the NAFTA renegotiations, a five-year sunset clause. At the time, Trudeau personally offered to come to the U.S. to work out the details of a revised trade deal, but Vice President Mike Pence called him just before his visit to say that the visit would only take place if Trudeau accepted the sunset clause. Trudeau refused and the visit was canceled. Neither President Trump nor Pena Nieto mentioned the sunset clause during their announcement.

Mr. Trump has derided NAFTA as the worst trade deal ever signed. But it's taken more than 18 months in office to reach any sort of agreement.