In the phone call on Sunday, Mr Kuczynski asked Mr Trump to "evaluate" the situation.
So far efforts to arrest Mr Toledo have been stifled by legal complications.
The US has said that it is unable to arrest the ex-leader until further information on the case against him has been shared, Peruvian officials say.
Authorities in Peru, who requested Mr Toledo`s arrest last week on charges of corruption, fear that he could try to fly to Israel.
However Israel has said that he would not be allowed to enter the country.
Mr Toledo is a visiting professor at Stanford University and his wife, Eliane Karp, has Israeli citizenship.
He governed Peru from 2001 to 2006 and is accused of receiving money from Brazilian building firm Odebrecht in return for a contract to build stretches of a highway linking the country with Brazil.
A reward of $30,000 (£24,000) has been offered for any information leading to Mr Toledo`s capture.
Israel`s foreign ministry said in a statement that Mr Toledo would not be allowed entry into the country until "his affairs in Peru are settled".
Odebrecht is at the centre of a multi-national corruption scandal. It admitted, as part of a plea deal with the US justice department, to paying nearly $800m (£640m) in bribes to governments across Latin America.
The company said it paid $29m in Peru to secure contracts between 2005 and 2014.
That time spans the presidencies of Mr Toledo and his two successors in office, Alan Garcia and Ollanta Humala, who have also denied any wrongdoing.
Peru said Interpol issued a red alert to 190 member countries to help find him, but Mr Toledo did not appear on its list of wanted persons.
The current president of Peru, Pedro Pablo Kuczynski, has urged him to return to the country to face questioning.
Meanwhile, Interior Minister Carlos Basombrio told Reuters news agency that the US had informed Peru that there did not appear to be sufficient probable cause to merit detaining Mr Toledo, and had asked the country to refile its request.
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