A possible starting date of 3 October has been given separately by various unnamed diplomatic sources.
EU leaders are still discussing dates at the two-day summit that began earlier on Thursday in Brussels.
The sources said that the aim of the talks - which could take up to 15 years - would be full membership, but Turkey's entry would not be guaranteed.
Turkish Prime Minister Recep Tayyip Erdogan also said he expected a deal with the EU on the issue of Cyprus before the end of the meeting.
Asked what the latest progress in discussions on the Cyprus issue was, Mr Erdogan told reporters in Brussels it "will be resolved tomorrow", without elaborating.
European Commission president Jose Manuel Barroso had insisted that the talks should be on full membership rather than any compromise measure.
Mr Barroso urged Turkey to "go the extra mile" and recognise Cyprus, which is an EU member.
He asked what kind of message Turkey was sending if it did not recognise all the members of the club it wanted to join.
But Turkey, which occupies northern Cyprus, has said it will not bow to demands to recognise the country, calling the issue a "red line".
'Emotional challenge'
There are widespread concerns over the impact on EU labour markets of Turkey's 71 million-strong population, which is predicted to rise to more than 80 million by 2015 and to overtake Germany's population soon after.
But Mr Barroso voiced enthusiasm for Turkey's bid to join the 25-nation European Union.
"The challenge for Turkey is to win the hearts and minds of those European citizens who are open to, but not yet fully convinced of, Turkey's European destiny," Mr Barroso said.
Mr Erdogan has cautiously welcomed EU leaders' backing for the Turkish bid.
Turkish newspaper on Thursday, Mr Erdogan promised to scrutinise "every word" of the EU leaders' decisions.
Turkish Foreign Minister Abdullah Gul has warned that imposing strict preconditions on talks would amount to another "red line" Turkey would not cross.
On Wednesday evening, French President Jacques Chirac spoke out in favour of Turkish membership in a live interview on French TV. But a poll by Le Figaro newspaper this week suggested that two-thirds of French people were against Turkey joining the EU.
On Wednesday, the European parliament voted strongly in favour of opening membership talks with Turkey.
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.
