The talks in Amman, at which Abbas sought assurances over Jerusalem's Al-Aqsa mosque compound that has been the focal point of unrest, came as a Palestinian tried to stab an Israeli security guard in the West Bank and was killed, according to police.
Kerry, who held talks with Israeli Prime Minister Benjamin Netanyahu in Berlin on Thursday, later went into a meeting with Jordan's King Abdullah II, custodian of Jerusalem's flashpoint Al-Aqsa mosque compound.
Kerry told reporters he was "hopeful", without elaborating, while Abbas said: "All the time we have the hope."
The latest wave of violence erupted over the status of the Al-Aqsa mosque compound, which is sacred to both Jews and Muslims and has long been a crucible for tensions fuelling the decades-old Israeli-Palestinian conflict.
Netanyahu has repeatedly accused Abbas of fanning the flames by suggesting Israel wants to change the status quo at the site under which Jews are allowed to visit but not pray.
"The king has guardianship over Al-Aqsa mosque and the holy places and will not allow this manipulation by Netanyahu," he told reporters after the Abbas-Kerry meeting.
The international community is seeking a halt to a wave of violence that many fear heralds a third Palestinian intifada, or uprising.
Kerry, his Russian counterpart Sergei Lavrov, EU foreign policy chief Federica Mogherini and UN Secretary General Ban Ki-moon -- members of the Middle East peacemaking Quartet -- appealed for "maximum restraint" after talks yesterday in Vienna.
Tensions over the holy site have triggered a wave of lone-wolf knife attacks, shootings and car-rammings against Israelis, as well as clashes between Palestinians and Israeli security forces.
Since the start of this month, 51 Palestinians and one Israeli Arab have died in clashes or while carrying out attacks.
Eight Israelis have been killed in attacks. One Israeli Jew and one Eritrean have been killed after being mistaken for attackers.
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