The Prime Minister made the call last night after joining other key European leaders to push for tighter sanctions on the Kremlin.
A Downing Street spokeswoman said Mr Cameron made clear to President Putin the downing of the plane was "totally unacceptable".
She added: "The evidence suggested pro-Russian separatists were responsible and the Prime Minister made clear that if Russia wants to put the blame elsewhere they would need to present compelling and credible evidence.
"The Prime Minister made clear that our priority is to get experts to the crash site so they can recover and repatriate the victims and collect any evidence necessary for the investigation."
Ukraine and pro-Russian rebels have accused each other of using a missile to shoot down the Boeing 777 on Thursday, killing 298 passengers - including Scottish oil worker Stephen Anderson, 44.
Tributes have poured in for the former RAF search-and-rescue officer. His employer, Maersk Drilling, said: "Our thoughts go out to Stephen Anderson's family and his colleagues on board Maersk Deliverer, and we will make every effort possible to support and assist family and colleagues."
The UK Government has said the "unavoidable conclusion" is that the plane was hit by a missile fired from rebel-held territory, "almost certainly a missile supplied by the Russians".
Mr Cameron, French President Francois Hollande and German Chancellor Angela Merkel have decided foreign ministers should announce fresh sanctions on Russia at a meeting of the EU's Foreign Affairs Council tomorrow.
A Downing Street spokesman said: "All three leaders agreed … that the EU must reconsider its approach to Russia and that foreign ministers should be ready to impose further sanctions on Russia when they meet on Tuesday."
The No 10 spokesman said the leaders agreed the immediate priority was getting access to the crash site and ensuring specialist teams so they could recover bodies and repatriate them.
They said Mr Putin "has an important role to play by persuading the separatists to grant access and to work with the international community to ensure that all that needs to be done can be done as soon as possible".
Ukraine's security service has released a recording it claims features pro-Russian separatists talking about trying to hide the black boxes from flight MH17 at the request of Moscow.
In the audio clip one fighter can be heard telling another: "Our friends from high above are very much interested in the fate of the 'black boxes'. I mean people from Moscow. Please, co-operate with the Ministry of Emergency. All that you find must not come into somebody else's hands. Try to take everything that you find so that it doesn't get into somebody else's hands. All those people that are coming, OSCE and so on."
Before the disaster, EU leaders agreed to sanction some Russian firms and block some loans to Russia, but the measures were less stringent than restrictions announced by the US.
The EU has been under pressure from the US and Ukraine to take a harder line on Russia .
Yesterday Foreign Secretary Philip Hammond warned that unless Moscow's position on the crash radically changed, Britain would push reluctant European states to agree new sanctions.
"Some of our European allies have been less enthusiastic and I hope that the shock of this incident will see them … more willing to take the actions which are necessary to bring home to the Russians that when you do this kind of thing it has consequences," he said.
Mr Hammond added: "The Russians have influence if not direct control over these people.
"They have been supplying them, they have been supporting them, … They cannot deny their responsibility for the acts these people are carrying out."