The fight against so-called Islamic State (IS) will dominate talks in Washington shortly between French President Francois Hollande and US President Barack Obama.
The Paris suicide attacks claimed by IS have prompted the US to issue a worldwide travel alertto its citizens.
President Hollande is meeting other world leaders this week, hoping to forge a stronger alliance against IS.
But Turkey’s downing of a Russian jet has fuelled tension over the Syria war.
Russia says a Sukhoi Su-24 warplane was shot down over Syria on Tuesday by an air-to-air missile fired by a Turkish F-16 fighter. Turkey says the Russian jet had entered its airspace.
As Turkey is a Nato member the incident has put a new strain on Russian-Western efforts to neutralise the IS threat in Syria and Iraq.
The US is unlikely to support French efforts to build a Russian-Western military alliance against IS, correspondents say. But there is some military co-ordination with Russia.
France is now sending warplanes from the aircraft carrier Charles de Gaulle, off the Syrian coast, to pound IS targets.
Russia has also stepped up air raids against anti-government rebels in Syria, since a Russian airliner was blown up over Egypt by suspected IS-linked militants last month.
But Western leaders have urged Russia to focus its raids on IS, not on other groups fighting Syrian President Bashar al-Assad.
French police are examining what appears to be a suicide bomb belt dumped on a Paris street following the 13 November attacks that killed 130 people.
It is said to resemble belts used by the attackers and was found in a suburb which a suspect is thought to have passed through after the attacks.
The key suspect, Salah Abdeslam, remains at large and a massive manhunt for him continues.
After the attacks he fled to Belgium, where he and some of the other jihadists were based.
Read more: BBC