Eid, which began Friday, is the holiday that follows on the Muslim fasting month of Ramadan.
Netanyahu's office said the premier also told Abbas in a telephone conversation "that the citizens of Israel want peace" and that "Israel would continue to act toward regional stability.
The last time the two leaders spoke was in June 2014.
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Netanyahu had asked Abbas for help in ensuring the safe return of three Israeli teens kidnapped by Palestinian militants, who murdered them.
The kidnapping touched off a series of events that led to a devastating 50-day war later in the Gaza Strip between the Mediterranean enclave's Islamist rulers Hamas and Israel.
US-backed peace talks between the Palestinians and Israel collapsed in April 2014 after nine months of fruitless meetings amid bitter recriminations and mutual blame.
Since the war, relations with the Palestinian Authority headed by Abbas have deteriorated further, with Palestinian moves against the Israelis in the international arena.