Russia is pushing for an international Syria conference and has already discussed the plan with Jordan as well as the European Union, Iran and Iraq.
Putin's strident rhetoric and a flat-out refusal to support sanctions against Moscow's Soviet-era ally Syria have pitted him against the West.
But Israeli president Shimon Peres urged the Russian strongman to play a bigger role in the region and specifically to use his influence with Tehran to dampen what Israel and much of the international community say are Iran's nuclear arms ambitions.
"I ask as a personal request that you make your voice heard against a nuclear Iran, against genocide," Peres told Putin after talks.
And Peres warned of "a real danger that Syrian chemical weapons will reach the hands of Hezbollah and Al-Qaeda; please act with urgency to stop that unacceptable situation."
"President Peres presented President Putin with the proposal that Syria will, for two years, be handed over to an Arab League mandate with UN support, until democratic elections can be held in the country," a statement from Peres's office added.
"I am confident that Russia, which defeated fascism, will not allow today's threats to continue. Not the Iranian threat. Not the bloodshed in Syria," Peres said earlier in the resort town of Netanya where he and Putin unveiled a World War II monument.
Netanyahu also warned that Iranian nuclear weapons pose "a severe danger, first and foremost to Israel, but also to the region and entire world."
Israel, which is widely believed to have the Middle East's sole if undeclared nuclear arsenal, has said Iran's programme poses an existential threat and has warned it reserves the right to use all means necessary to respond, including military.
The international community has been pursuing talks with Tehran, but three high-level meetings -- the most recent in Moscow -- have failed to produce any breakthrough.
Putin stressed that all disputes should be resolved on the basis of international law. "We think it is unacceptable according to international law to think of mutual destruction, regardless of who is voicing such threats," he said in comments translated into Hebrew.
The P5+1 group -- Britain, China, France, Russia and the United States plus Germany -- has agreed to another round of discussions in Istanbul on July 3, but Israel has warned that lengthy talks give Iran time to continue uranium enrichment.
Earlier on Monday, Putin sought to highlight common history that he said brings the two nations closer.
He said the Netanya monument would serve "as a reminder that the world is still fragile and we should do our best in order for the criminal Nazi doctrines, no matter what form or shape they take, to be left in the past."
A private visit to the Old City of Jerusalem, where Putin was to have visited the Holy Sepulchre church, the traditional site of Christ's tomb, and the Western Wall, sacred to Judaism, was cancelled at the last minute, according to photographers waiting at the sites.
On Tuesday, Putin is to meet Palestinian president Mahmud Abbas in the West Bank city of Bethlehem for talks likely to focus on the stalled peace process. Later Putin will travel to Jordan.
The peacemaking Quartet, which groups Russia, the United States, European Union and United Nations, has been trying to nudge the two sides back to direct talks, on hold since late 2010.
But little progress has been made, and the Palestinians have insisted on a settlement freeze before talks resume, while Israel calls for new discussions without preconditions.