Mr Kuczynski has reportedly lost the support of key members of his cabinet.
Scandal-plagued Odebrecht says it paid Mr Kuczynski nearly $5m (£3.7m) while he was economy minister and head of cabinet. He denies wrongdoing.
Odebrecht was fined a record $3.5bn by the US last year for bribing officials from various countries.
Fraudulent “advisory fees” was one method the firm used to funnel bribes to officials, according to the case against the firm.
Former company president Marcelo Odebrecht, who is serving a 19-year jail sentence in Brazil and has agreed to co-operate in return for leniency, said Mr Kuczynski was hired as a consultant in Peru a decade ago.
He also told prosecutors that the construction company had paid $29m (£22m) in bribes to Peruvian officials over many years.
Kuczynski, 79, initially denied receiving any money from Odebrecht. He now denies receiving any illegal payments but admits working on an advisory basis for the firm.
Peru’s right-wing opposition Popular Force party, which controls the country’s congress, claims there is enough evidence to prove corruption by Mr Kuczynski, a centrist elected last year to a five-year term.
The party has warned that it will begin impeachment proceedings against him if he does not offer his resignation on Thursday.
“It’s obvious that him staying on in the nation’s highest office is untenable,” said party spokesman Daniel Salaverry.
Mr Kuczynski, a former Wall Street banker, has pledged to testify before a congressional commission on Odebrecht wrongdoing in Peru on December 22.
On Wednesday, Ecuador’s vice president, Jorge Glas, was sentenced to six years for taking $13.5 million in bribes from Odebrecht, becoming the highest-ranking politician to be convicted in the scandal. (Courtesy BBC)