Britain offered its assistance to Russia two weeks after Turkey downed a Russian warplane in Syria, leading to a severe crisis in ties with Moscow.
On Wednesday Turkish Prime Minister Ahmet Davutoglu said his country was ready to work with Moscow to prevent a repeat of the November 24 incident but also accused Moscow of pursuing a campaign of "ethnic cleansing" in Syria.
"Vladimir Putin invited British specialists to take part in decoding flight data recorders of the downed Su-24," the Kremlin said after the phone talks with Cameron.
Cameron's office said for its part that the prime minister would "consider President Putin’s request to send British experts to assist the investigation."
A Cameron spokesman told reporters earlier in the day: "We're always happy to help and assist other countries in issues of security."
Russia has recovered the black box recording flight data from the fighter jet and Defence Minister Sergei Shoigu showed the orange-painted box to Putin during a Kremlin meeting on Tuesday evening.
Putin said at the Kremlin meeting that it should be analysed with the involvement of foreign experts to determine the plane's flight path and position, points on which Moscow and Ankara furiously disagree.
Cameron, whose country last week launched its own bombing campaign in Syria, expressed condolences over the downing, the Kremlin said in a statement.
"It has been noted that Russia and Britain have similar approaches regarding the threat from ISIL and other terrorist groups operating in the region," the Kremlin said, referring to the Islamic State (IS) group.
The two leaders also discussed ramping up cooperation "in this context", the statement said without elaborating.
- 'Tackling Daesh together' -
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Cameron's office struck a similar note, saying Cameron called Putin to update him on Britain's Syria campaign.
"The prime minister and president agreed that the UK and Russia should work together, with other international partners, to tackle Daesh and the threat it poses, and on the political process to bring peace to Syria," it said, using the Arabic acronym for IS.
Russia has been pushing for establishing a broad anti-IS coalition since launching its controversial bombing campaign in Syria in September.
After the November 13 Paris attacks Putin hosted French President Francois Hollande in Moscow, with the two agreeing to increase cooperation on Syria and share intelligence.
The Kremlin however has lamented that global powers such as the United States, which has been bombing IS targets in Syria together with its allies for more than a year, are not ready to form a coalition with Russia.
Relations between Moscow and London have been tense for years.
Russia recently criticised Britain for suggesting that a bomb caused the crash of a Russian passenger plane over Sinai in October before the investigation was completed.
- 'Ethnic cleansing' -
The downing of the jet has led to a severe crisis in ties between Moscow and Ankara, with Russia hitting Turkey with a raft of sanctions.
Turkish Prime Minister Ahmet Davutoglu said Wednesday he was "ready to work" with Russia to prevent a repeat of last month's incident.
But he also accused Russia of pursuing a campaign of "ethnic cleansing" in Syria and strengthening IS.
"Russia is trying to make ethnic cleansing in the northern Latakia (region) to force (out) all Turkmen and Sunni populations who do not have good relations with the (Syrian) regime," Davutoglu told foreign reporters in Istanbul.
Putin and Cameron spoke as a broad range of Syrian opposition groups, including armed factions, held unprecedented talks in the Saudi capital on forming a united front for talks with Bashar al-Assad.
Russia said Wednesday that Moscow, Washington and the United Nations would hold Syria talks in Geneva on Friday in an effort to end a war that has killed more than 250,000 people and forced millions from their homes since March 2011.