"I am not going to abdicate my honor, my values or my responsibilities as president," Kuczynski said in a televised speech to the nation, backed by his ministers.
"I won't run, I won't hide nor do I have any reason to do so," he said, promising to cooperate with investigations by Congress and the country's attorney general's office.
The right-wing Popular Force party, which controls the Congress, had warned it would begin impeachment proceedings against him if he did not resign by Thursday.
Odebrecht, a Brazilian engineering and construction giant at the center of multiple corruption cases throughout Latin America, revealed on Wednesday it had paid Kuczynski five million dollars in consulting fees between 2004 and 2013.
For part of that period, Kuczynski was economy minister and head of cabinet for then-president Alejandro Toledo, whom Odebrecht said it paid $20 million in kickbacks to win a contract managing a freeway.
"It's obvious that him staying on in the nation's highest office is untenable," Popular Force spokesman Daniel Salaverry said, referring to Kuczynski, a centrist who took office last year after being elected to a five-year term.
But in his speech, Kuczynski said he would not step down because "it cost us a lot to recover democracy, and we are not going to lose it."
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