David Cameron is meeting Vladimir Putin as Western allies try to persuade the Russian president to co-operate in the international struggle against terror group Islamic State in the wake of attacks in Paris and Egypt.
The meeting on the fringe of the G20 summit in Turkey comes after Mr Putin and US president Barack Obama met on Sunday for talks on the civil war in Syria which US officials characterised as "constructive".
Downing Street sources said the Prime Minister hopes to reassure Mr Putin that Russia's interests will be "protected" in the transition to a new settlement in Syria after the departure of president Bashar Assad, a close ally of Moscow.
Downing Street sources said Mr Cameron continues to believe there is no role for Assad in a future administration in Damascus.
But he wants to use the talks to explore how Russia and the West can "bridge our differences" over Syria by recognising the shared threat they face from IS and the shared interest they have in finding a way forward.
The West hopes that last month's downing of a Russian airliner over the Sinai - which Britain believes was caused by an IS bomb - will help persuade Mr Putin to shift his stance on Syria and turn his firepower against the terror group which holds large swathes of the country.
Mr Putin claims the campaign of airstrikes he launched in September is directed at "terrorists", but the West accuses him of instead targeting the forces of the moderate opposition ranged against Assad.
The summit of 20 leading world economies at the Mediterranean resort of Antalya - just 300 miles from the Syrian border - has been dominated by the fight against terror, following Friday's slaughter of 129 people in Paris and the death of 224 on board the Russian Metrojet Airbus in Egypt last month.
Following Sunday's unscheduled 35-minute talks between the American and Russian leaders, a US official said: "President Obama and President Putin agreed on the need for a Syrian-led and Syrian-owned political transition, which would be proceeded by UN-mediated negotiations between the Syrian opposition and regime as well a ceasefire."
Speaking ahead of today's meeting, Mr Cameron said: "It's become even more clear that our safety and security depends on degrading and ultimately destroying Isil whether it's in Iraq or Syria.
"We're playing a huge role in that already in Iraq. Others are taking action in Syria which we both support and enable, but we've got to keep on making the case that we will be safer in the UK, in France, right across Europe if we destroy this death cult once and for all.
"We have our differences with the Russians, not least because they've done so much to degrade the non-Isil opposition to Assad, people who could be part of the future of Syria.
"But the conversation I want to have with Vladimir Putin is to say 'Look, there is one thing we agree about which is we'd be safer in Russia, we'd be safer in Britain if we destroy Isil. That's what we should be focusing on'."
Mr Cameron's talks with Mr Putin will be followed by a meeting of the Quint - an informal group of Western powers within the G20, made up of the UK, US, France, Germany and Italy - to assess progress and discuss how further efforts on Syria can be co-ordinated.
And the leaders will observe the international minute's silence for Paris at 11am UK time.
It will be the first time the PM and Russian President have spoken face to face since the Brisbane G20 summit in November last year, when frosty discussions were dominated by events in Ukraine.
Mr Cameron will also urge Mr Putin, from a humanitarian point of view, to put pressure on the Assad regime to stop using barrel bombs against its citizens - something which is believed to have driven a recent surge in refugees fleeing the country.
The PM hopes to revive the spirit of lengthy discussions at a 2012 meeting at Putin's dacha in Sochi, when the two men were able to have "deeper" conversations about Syria's future.
But sources stressed that the PM was "realistic" about the prospects for swift progress and acknowledged there was "a lot still to be worked out" before proposals for a settlement for Syria can be agreed.
A meeting of foreign ministers in Vienna on Saturday agreed a two-year timeline towards the election of a new Syrian government, but key details remained unresolved, including what part Assad might play in any new settlement and a decision on which opposition groups should be shunned as terrorists.
At a working dinner on Sunday evening, Mr Cameron pushed for a concerted effort to improve airport security around the world.
G20 states should be ready to help poorer countries financially to raise their security standards, and to assist those - like Tunisia - whose tourist industries have suffered a slump in business as a result of terrorism, he said.