The right-wing Popular Force party, which controls Peru's Congress, deems there is sufficient evidence to show "acts of corruption" by Kuczynski, a centrist who took office last year for a five-year term.
It has given him a deadline of Thursday to resign, or it would start impeachment proceedings.
"It's obvious that him staying on in the nation's highest office is untenable," said the spokesman for the party's lawmakers, Daniel Salaverry.
The accusation against the president stems from Odebrecht's revelation on Wednesday that it paid Kuczynski nearly $5 million to advise companies tied to it between 2004 and 2013.
For part of that period, Kuczynski was economy minister and head of cabinet for then-leader Alejandro Toledo, whom Odebrecht said it paid $20 million in kickbacks to win a contract managing a freeway.
Following an investigation by US authorities, Odebrecht agreed a year ago to pay a record $3.5 billion fine after admitting to paying $788 million in bribes across 12 countries to secure public works contracts.
Fictitious "advisory fees" were one scheme the Brazilian company used to funnel bribes to officials.
'Regional crisis'
The Odebrecht scandal has ensnared politicians in several other countries, including Mexico, Panama, Venezuela, and of course in Brazil itself.
It has generated "a regional crisis that shows that corruption is structural in all those countries," the former head of the anti-corruption watchdog Transparency International, Jose Ugaz, told AFP.
On Wednesday, Ecuador's vice president, Jorge Glas, was sentenced to six years in prison for taking what prosecutors said was $13.5 million in bribes from Odebrecht. He is expected to appeal the conviction and sentence.
Glas so far is the biggest political scalp from the Odebrecht affair. But Kuczynski risks outranking him if Peru's opposition gets its way.
"What the Peruvian people demand is that the president steps down and that there is a constitutional transition so that the vice president of the republic assumes Peru's presidency," Salaverry said.
Kuczynski, a 79-year-old former Wall Street banker, had denied that he received any money from Odebrecht. But on Saturday, he admitted he had provided advisory services, but did not give details. He denies receiving any illegal payments.
The president has said he will testify before a congressional commission looking into the extent of Odebrecht wrongdoing in Peru on December 22.
In the event he does step down, Peru's First Vice President Martin Vizcarra -- currently ambassador to Canada -- would replace him.
The crisis is benefiting the Popular Force party, which hold 71 of the 130 seats in Congress.
It is led by Keiko Fujimori, daughter of disgraced former president Alberto Fujimori, in prison for murder, kidnapping and other human rights abuses as well as corruption during his 1990-2000 reign.
Keiko Fujimori was narrowly beaten by Kuczynski in the 2016 presidential election.
The political crisis was blowing up a month before a visit to Peru by Pope Francis.
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