Peruvian police launched a manhunt for him after a judge ordered his arrest Thursday night.
The net appeared to be closing on the former Peruvian president as Israel said it would refuse entry to Toledo, who is accused of taking bribes from Brazilian construction giant Odebrecht.
Toledo's French-born wife Eliane Karp has Israeli citizenship.
"President Kuczynski has asked Donald Trump to evaluate... with the State Department the idea of sending Toledo back to Peru," where he is wanted on charges of taking $20 million in bribes while in office, state news agency Andina reported on Sunday.
However, the United States said it could not arrest Toledo until it received more information on the case, according to Peruvian officials, who were scrambling to send investigation documents to their US counterparts.
Peruvian authorities believe Toledo, 70, is in San Francisco. He is a visiting professor at nearby Stanford University.
Toledo issued a statement on Twitter late Sunday night, without revealing his whereabouts but denying he was on the run as he had not been facing charges when he left Peru.
"The court in Lima did not request my testimony to help with their investigation. On the contrary, it accused me of crimes that I have not committed and that the court cannot prove," he said.
Toledo has branded the accusations against him as political persecution. But he has struggled to explain the source of the money.
A White House readout of a call between Kuczynski and Trump did not mention Toledo.
Instead, it said the pair discussed the "humanitarian situation" and other concerns in Venezuela, and "the need to promote democratic governance throughout the Western Hemisphere."
The White House hinted at a future meeting between the two leaders.
Toledo, once hailed as an anti-corruption champion, is accused of taking a huge payoff from Odebrecht in exchange for a massive highway contract connecting Peru and Brazil during his 2001-2006 presidency.
Peruvian police launched a manhunt for him after a judge ordered his arrest Thursday night. There is a $30,000 reward for information leading to his capture.
The Peruvian government has indicated that Toledo might try to flee to Israel.
"We heard from a solid source that he was trying to flee to Israel. So we alerted Israel," Interior Minister Carlos Basombrio said on RPP Radio.
But Israel blocked any such effort by Toledo.
"Toledo will be allowed in Israel only when his affairs in Peru are settled," foreign ministry spokesman Emmanuel Nahshon said in a statement.
Peru's authorities have a warrant for Toledo to be detained and held in preventive custody for 18 months pending a full investigation.
He is accused of influence peddling and money laundering.
At the moment, Toledo could leave the United States, since the US has not yet received the documents on which the warrant is based, according to Basombrio.
Israel's foreign ministry later said he was not aboard a flight from San Francisco that landed at the country's main international airport late Sunday.