Ankara is also haggling for a refugee swap under which the European Union would resettle one Syrian refugee from Turkey in exchange for every Syrian refugee that Turkey takes back from the overstretched Greek islands.
Under the last-minute proposals by Turkish Prime Minister Ahmet Davutoglu, the 28-nation bloc would also bring forward visa-free travel for Turks to June, and speed up its EU membership bid.
The EU is paying an increasingly high price to secure Turkey's help in dealing with the biggest migration crisis since World War II, but has little choice as Turkey is the main launching point for the Greek islands.
"It's been four months. They are yet to deliver," Erdogan said in a speech in Ankara. "Mr prime minister is currently in Brussels. I hope he will return with the money."
More than one million refugees and migrants have arrived in Europe since the start of 2015 -- the majority fleeing the war in Syria -- with nearly 4,000 dying while crossing the Mediterranean.
In a surprise move, Davutoglu unveiled what his spokesman called a "new proposal" at a lunch with EU leaders in Brussels, forcing them to extend the lunchtime summit until dinner to discuss his demands.
European Parliament head Martin Schulz confirmed Turkey's demand for an extra three billion for 2018 on top of the 2016-17 money, saying it "will require additional (EU) budgetary procedures".
One EU diplomat told AFP Turkey was proposing "a potential gamechanger" where it will take back not only irregular economic migrants who have reached the Greek islands but also those from Syria deemed genuine refugees.
"In return, we have said for every Syrian they take back, we will resettle one Syrian" from camps in Turkey, where 2.7 million Syrian refugees are living, the diplomat added.
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