Turkey expects to receive money and political concessions in return for clamping down on its borders and keeping refugees in the country.
Mr Davutoglu hoped for a “historic day” in Turkey’s relations with the EU.
Correspondents say there is still disagreement about how much each EU member state will give Turkey.
Almost 900,000 migrants have made the trip to Europe this year.
Many are fleeing conflict in Syria, Iraq and Afghanistan, and lived in makeshift camps in Turkey before their journey.
But it is understood that the EU is offering Turkey €3bn ($3.2bn; £0.7bn) over two years towards tightening border controls and improving conditions for those large numbers of migrants and refugees still within its borders.
However, it has still not been decided how much each EU member state will contribute or how the money will be spent.
Austrian Chancellor Werner Faymann said he was optimistic a deal could be reached.
“Turkey now wants €3bn a year to invest the money in schools and accommodation. We will meet somewhere in the middle,” he said, quoted by Reuters.