LIMA, Peru — A group of lawmakers on Friday threatened to oust President Pedro Pablo Kuczynski of Peru, the latest leader accused of involvement in a corruption scandal that has shaken some of Latin America’s biggest countries.

The dispute involves Odebrecht, a Brazilian construction giant, which last year admitted to paying roughly $800 million in payoffs in exchange for lucrative projects. That revelation set off a flurry of investigations by prosecutors and lawmakers, principally in Latin America, seeking to learn who was on the receiving end of the payments.

According to documents sent to Congress and released this week, Odebrecht paid $782,000 in advisory fees to Westfield Capital, a company Mr. Kuczynski owns. Most of the payments occurred between 2004 and 2007, while Mr. Kuczynski served as Peru’s economy minister and prime minister.

On Friday, top politicians from the country’s governing parties signed a document calling Mr. Kuczynski “morally handicapped” and declaring the president’s office to be “vacant.” It set the stage for a vote in Congress that would give Mr. Kuczynski three days to prepare a defense before lawmakers decided his fate, as early as next week, experts said.

Mr. Kucynski’s opponents would need a two-thirds majority to remove him.

“By Christmas we will probably have a new president,” said Eduardo Dargent, a political-science professor at the Pontifical Catholic University of Peru, noting that Mr. Kucynski’s party holds only a few seats in Congress to defend him. Unlike impeachment proceedings, the ouster proposed by lawmakers would not require a lengthy trial, he said.

Mr. Kuczynski, 79, has refused to step down. On Thursday night the president gave a speech in which he said he had done nothing wrong, but he did not deny that payments were made. “I’m not running, I’m not hiding, I have no reason to do so,” he said.

Mr. Kuczynski said he had invoices for the transactions, adding that while Westfield Capital did belong to him, he had not managed it personally. He said he had reported the transactions to the authorities in January and had not signed any contracts with Odebrecht.

“I am an honest man and have been all my life,” he added.

Mr. Kuczynski’s center-right Peruvians for Change party holds only 18 of the legislature’s 130 seats. His main rival, Keiko Fujimori, lost to him in the last presidential election, in June 2016, but maintains a large majority in the legislature through her right-wing Popular Force party.

In September, the opposition party flexed its muscles by staging a no-confidence vote against the president, and calling for the entire cabinet to resign and for Mr. Kuczynski to pick new ministers.

“The president increasingly appears unlikely to serve out his term in office,” wrote Fernando Freijedo, an analyst with the Economist Intelligence Unit, in a note to clients on Friday.

Mr. Kuczynski is the latest in a growing list of leaders who have been accused or sentenced to crimes related to the scandal.

On Wednesday, a court in Ecuador sentenced the country’s vice president, Jorge Glas, to a six-year prison sentence on charges of receiving millions of dollars in bribes from Odebrecht through his uncle.

In October, an ousted Brazilian prosecutor released a tape in which an Odebrecht official said that President Nicolás Maduro of Venezuela had accepted millions in campaign contributions in exchange for contracts.

Two former Peruvian presidents also face charges of accepting bribes, with one in jail and another at large.

Prosecutors have said Odebrecht paid bribes on 100 projects in more than a dozen countries, from Mexico to Angola, in one case buying a local bank branch to hide the transactions, and even opening a division dedicated to payoffs.

Throughout Latin America, the company built bridges, dams, power plants and roads, along with a highway linking Brazil and Peru that greatly exceeded its budget. Nearly three years of investigations have prompted at least 77 Brazilian Odebrecht executives to sign plea deals, and led to the imprisonment of the company’s former chief executive, Marcelo Odebrecht.