Netanyahu's phone call to Abbas was their first direct political contact since 2012, and took place as Israel carried out a vast search operation in the West Bank to find the three youths, whom the Israeli premier said were kidnapped by Hamas militants last week.
As the manhunt entered its fourth day, Israeli troops arrested scores of Hamas members, among them MPs and former ministers, bringing the total number of Palestinians rounded up since Friday to 150, the biggest sweep in the territories in years.
"The Hamas kidnappers went out from territory controlled by the Palestinian Authority and returned to territory controlled by the Palestinian Authority," he said.
So far, there has been no formal claim of responsibility, and Hamas has dismissed Israel's accusations as "stupid".
Abbas's office issued a statement condemning "the kidnapping of three Israeli youths".
But it also denounced Israel's "raids on Palestinian homes" and arrests of "many innocents" during which a 19-year-old Palestinian was killed.
Ahmad Arafat Sabarin, was shot dead during clashes in Jalazoun refugee camp near Ramallah which erupted when troops arrived on an arrest mission, medics told AFP.
Although arrests have taken place across the West Bank, troops are focusing their search on the southern city of Hebron and the surrounding area, home to some 663,000 Palestinians.
Overnight, soldiers arrested more than more than 40 Palestinians, the army said, including parliament speaker Aziz Dweik, who is a Hamas member.
New York-based watchdog Human Rights Watch urged Israel to refrain from "arbitrary" arrests in their search for the teenagers.
"Israeli forces should respect the laws of war with respect to the Palestinian population in the occupied territory and not carry out mass, arbitrary arrests," it said.
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