The EU has offered to begin membership talks with Turkey next year, with 3 October given as a start date.
Mr Erdogan is upbeat about Turkey's EU prospects
EU leaders said the aim of the talks - which could take up to 15 years - would be full membership, but Turkey's entry could not be guaranteed.
Discussion between EU leaders on finalising the offer resumes at the two-day summit on Friday morning.
EU leaders warned Turkey that it would have to take steps to recognise Cyprus before the talks started.
The EU has also announced that it will start accession talks with Croatia in April 2005.
However, talks will only begin if the country co-operates fully with the UN war crimes tribunal for the former Yugoslavia.
A draft statement circulated on Friday morning said Ankara would on Friday have to initial an accord extending an existing pact with the EU to the 10 new member states, which include Cyprus, Reuters news agency reported.
Italian Prime Minister Silvio Berlusconi said the 3 October date would give Turkey enough time to allow its parliament to ratify the move.
Turkey has not yet responded publicly to the offer, though Prime Minister Recep Tayyip Erdogan held talks overnight with Dutch Prime Minister Jan Peter Balkenende, who currently holds the EU presidency.
"We talked ... in detail about several aspects and tomorrow
(Friday) we will continue our conversation," Mr Balkenende said.
Mr Erdogan said earlier he expected a deal on the divided island before the summit ended on Friday.
Turkey, which occupies northern Cyprus, had originally said it would not bow to demands to recognise the country, calling the issue a "red line".
The internationally recognised southern part of Cyprus is an EU member.
Mr Balkenende said summit leaders still had to discuss conditions to be attached to the eventual accession.
TURKEY'S RED LINES
Negotiations must have Turkey's complete membership as the final aim
Turkey must not be forced to extend diplomatic recognition to the Republic of Cyprus
The decision to start talks must not be conditional on later decisions by EU leaders
There should be no special conditions imposed permanently on Turkey
The BBC's William Horsley in Brussels says doubts voiced by France and Austria about Turkey's accession have led to an offer that is hedged by strict conditions and falls short of a promise of eventual membership.
The Turkish Prime Minister, Recep Tayyip Erdogan, held talks overnight with EU officials to try to win more favourable terms.
But European Commission President Jose Manuel Barroso hailed the offer.
"Tonight the EU has opened its door to Turkey," he said.
"In doing so it is making a balanced offer which takes into
consideration the legitimate preoccupations of Turkey and legitimate
preoccupations also of EU member states."
"We believe this is an offer Turkey should be glad to
Mr Erdogan has promised to scrutinise "every word" of the EU leaders' decisions.
But our correspondent says Turkey faces a difficult choice - to reject the offer as too half-hearted, or accept it even though the terms are much less inviting that its government had hoped.
If Turkey's application is successful, the EU's frontier would extend deep into the Middle East.
It could become the first EU member with a majority Muslim population.