The European Union should "reconsider its approach to Russia" in light of evidence pro-Moscow separatists brought down flight MH17 in Ukraine, Downing Street said.
The Prime Minister spoke to Dutch counterpart Mark Rutte about the tragedy and the two men agreed that the 28-member bloc's relationship with Moscow should be reviewed.
Russia's ambassador to the UK Alexander Yakovenko has been summoned to the Foreign Office to be told that president Vladimir Putin must use his influence on the separatists to ensure access to the crash site, No 10 said.
The latest political developments came as the last two of the 10 UK victims who died aboard flight MH17 were named in a Malaysia Airlines passenger list as John Allen and Andrew Hoare.
The European Union has already imposed sanctions, including visa bans and travel freezes, against Russians involved in efforts to destabilise Ukraine, but the comments from Downing Street indicate that further action could be taken.
Foreign Secretary Philip Hammond warned Mr Putin that "the world's eyes are on Russia" and expressed his frustration with the Kremlin after chairing a high-level meeting in Whitehall.
He said: "We're not getting enough support from the Russians, we're not seeing Russia using their influence effectively enough to get the separatists, who are in control of the site, to allow the access that we need.
"This has brought the whole international community together.
"This is not about Russia and the West, this is about the whole community demanding that the proper access is made available to this site, the victims are properly recovered and evidence is secured.
"The world's eyes will be on Russia to see if she delivers on her obligations in the next couple of hours."
A Downing Street spokeswoman said monitors from the Organisation for Security and Co-operation in Europe had been given only "limited" access to the crash site in eastern Ukraine.
As well as Mr Rutte, the Prime Minister spoke to Australia's prime minister Tony Abbott about the catastrophe and they agreed to increase pressure at the UN Security Council for investigators to access the crash site.
The No 10 spokeswoman said: "All three leaders are clear that president Putin needs to actively engage with the international community and use his influence on the separatists to ensure they allow access to the crash site.
"The FCO in London has called in the Russian Ambassador to make these points.
"The PM also agreed with PM Abbott that the UK and Australia should work together to secure further pressure at the UN Security Council for swift and unhindered access to the crash site.
"The PM and PM Rutte agreed that the EU will need to reconsider its approach to Russia in light of evidence that pro-Russian separatists brought down the plane."
A team of six investigators from the Air Accidents Investigation Branch are in Ukraine, working with international counterparts on the next steps in the effort to establish what happened to flight MH17.
Experts from the Metropolitan Police are due to arrive in Ukraine tomorrow to assist in the grim task of recovering, identifying and repatriating the bodies of those killed.
Malaysia Airlines identified the nationalities of 298 people who were on board the doomed flight which includes 10 people from the UK, one of whom has dual UK and South African citizenship.
There are also 193 victims from the Netherlands, 43 from Malaysia including 15 crew and two infants and another 27 from Australia. The dead also include 12 people from Indonesia including one infant, four people from Germany, four people from Belgium, three from the Philippines plus a Canadian and a New Zealander.
Mr Allen was described as a "much-loved colleague" by his friends at international law firm NautaDutilh, who said they were "shocked" by his death alongside his wife and their sons.
Young holidaymakers, students, scientists and some entire families were on board the doomed flight from Amsterdam to Kuala Lumpur.
The bodies of passengers, which have been strewn across the crash site for nearly 48 hours, are finally being recovered by Ukrainian authorities.
Mr Hammond said the UK's priority is to ensure the victims are treated with dignity and respect as they are recovered from the scene.
Reporters said the bodies were being carried out on stretchers after a makeshift cordon was set up this morning, following a deal between separatists and Kiev.
Until now, many of the dead had been left uncovered, lying among the plane wreckage in an open field in the rebel-controlled area.
Ukraine has accused the rebels of already removing 38 bodies from the scene and taking them to the rebel-held city of Donetsk.
Kiev also claimed Russia had helped separatists destroy evidence at the crash site, a charge the rebels deny.
As dozens of victims' bodies lay in bags by the side of the road baking in the summer heat, international monitors at the scene said they were still being hampered by heavily armed rebels.
OSCE spokesman Michael Bociurkiw said: "Some of the body bags are open and the damage to the corpses is very, very bad - it is very difficult to look at."
He said the 24-member delegation was given further access to the crash site today but their movements were being limited by the rebels.
Malaysia Airlines announced that a 212-strong team, made up of personnel from various government and media bodies and its staff in Kiev and Amsterdam, had been detailed to help.
Five members will join Malaysia's search-and-recovery mission at the crash site in the Donetsk region, while 80 people will be stationed in Amsterdam to help grieving families.
Malaysia Airlines said it will no longer use the flight number MH17 in respect to those who died. The route will be called MH19 from July 25.
Among the British victims were Newcastle United fans John Alder, 63, and Liam Sweeney, 28, who were travelling to New Zealand to watch the team's pre-season tour.
Also on board were Glenn Thomas, 49, a press officer at the World Health Organisation and a former BBC journalist, Loughborough University student Ben Pocock, reportedly 20, and Leeds University student Richard Mayne, 20, from Leicestershire.
Helicopter rescue pilot and father-of-two Cameron Dalziel, 43, who is understood to be South African but was travelling on a British passport, also died along with Stephen Anderson, 44, a former RAF search and rescue co-ordinator.
Father-of-two Robert Ayley, who was originally from Guildford, Surrey, but was living in New Zealand, was also among the dead.