Russian President Vladimir Putin and his Palestinian counterpart on Monday discussed the escalation of tensions in the West Bank and East Jerusalem, a Kremlin statement said.
In a phone call initiated by Abbas, the statement said, the two leaders expressed hope that the situation will not turn into a large-scale Palestinian-Israeli confrontation.
"The importance of restarting the direct dialogue between Palestine and Israel, in particular, under the auspices of the Quartet of international mediators, was stressed," it added.
Established in Madrid in 2002, the Quartet on the Middle East comprises the UN, EU, US and Russia.
Putin and Abbas also discussed bilateral cooperation in trade and economy, and affirmed mutual commitment for further development of ties, the statement said.
The Russian president also discussed the situation in the West Bank and East Jerusalem with his Algerian counterpart Abdelmadjid Tebboune.
In a statement, the Kremlin said Putin and Tebboune stressed the importance of intensifying international efforts for the Middle East settlement.
Tensions have been running high across the Palestinian territories since the beginning of April amid repeated Israeli arrest campaigns in the West Bank.
On Friday, dozens of Palestinians were injured in clashes with Israeli forces in the flashpoint Al-Aqsa Mosque compound in Jerusalem.
Al-Aqsa Mosque is the world's third-holiest site for Muslims. Jews call the area the "Temple Mount," claiming it was the site of two Jewish temples in ancient times.
Israel occupied East Jerusalem, where Al-Aqsa is located, during the 1967 Arab-Israeli war. It annexed the entire city in 1980, in a move never recognized by the international community.