Turkey has demanded Europe's leaders hand over an additional £2.3 billion as the price of helping to stem the flow of refugees heading across the Aegean Sea.
At talks in Brussels, Turkish prime minister Ahmet Davutoglu also called for the speeding up of Ankara's EU membership application and quicker access to visa-free travel for Turkish nationals.
In the course of the meeting, David Cameron and other EU leaders did raise concerns about the Turkish government's latest press crackdown.
However misgivings over the weekend raid on the offices of the Zaman opposition newspaper in Istanbul were largely overshadowed by the pressing need for EU leaders to secure Turkish support in dealing with the migrant crisis.
The president of the European Parliament Martin Schulz confirmed that Mr Davutoglu had asked for an additional three billion euro (£2.3 billion) on top of the 3 billion euro already promised.
As the talks were extended into the evening, Mr Schulz said the Turkish requests for further financial support "are in the debate, are in the discussion".
British sources indicated the Government would be sympathetic to calls from Ankara for more help to support refugees from the Syrian civil war in the region - provided it could be shown that it was proving effective.
"If the system is working we will look at that. We will not turn the tap off," one source said.
Despite the need for Turkish co-operation on the migrant crisis, Downing Street said that Mr Cameron and other leaders had raised the issue of press freedom during a working lunch with Mr Davutoglu.
"The PM underlined the importance of protections for a free press and human rights in Turkey," a spokesman said.
Earlier, Mr Cameron insisted there was "no prospect" of Britain joining a common EU asylum system as officials in Brussels continued to grapple with the chaos caused by the huge influx into the continent.
The Financial Times reported that the European Commission was preparing to bring forward proposals at a further EU summit later this month to centralise control of asylum claims.
It would mean replacing the current Dublin regulation under which refugees have to claim asylum in the first country they arrive in.
That puts pressure on countries such as Greece and Italy - where many of the migrants from Syria and north Africa first set foot in the EU - while protecting those further away.
However Mr Cameron said that Britain's opt-out from the Schengen agreement meant that it would be unaffected by any changes.
"We have an absolutely rock-solid opt-out from these things so there is no prospect of Britain joining a common asylum process in Europe," he told reporters.
"We will have our own asylum approach, our own way of doing things, keeping our borders. It underlines the best of both worlds, the special status that we have."
The UK has announced plans to deploy a Royal Navy ship to join a Nato operation to tackle people smugglers bringing migrants across the Aegean Sea.
Defence Secretary Michael Fallon said the amphibious landing ship RFA Mounts Bay will use an onboard helicopter to provide information for the Turkish coastguard about the routes being used by smuggling gangs, giving them a better chance of intercepting boats attempting the perilous crossing to Greece.
"What's essential is to work out where this people smuggling is done from and then to get a policy in place of returning people, which in the end will stop people making this very dangerous crossing," he told BBC Radio 4's Today programme.
"We have to put a stop to this because otherwise we are going to see more lives lost and more misery."
Around 1,800 migrants a day arrived in Greece in February, with more than 116,000 migrant arrivals across the Aegean already this year.