The PM wants to move the UK's exit date from Friday to 30 June - but leaders are split over the length of delay.
Seventeen member states so far favour a long extension, including Germany, while France wants a short delay.
Reuters reports that the French president told the summit a delay beyond 30 June would jeopardise the EU.
A no-deal Brexit was not the worst option and the EU would run a higher risk if the UK obstructed EU operations, a French official is reported to have said.
The PM spoke for about an hour at the summit, before leaving the room to leave the EU leaders to discuss her request.
All the leaders have spoken and had a break, and they will now each speak for a second time.
The EU member states have to reach a unanimous decision by law.
While 17 EU leaders in favour of longer #Brexit extension, France is refusing to countenance anything beyond 30 June at the moment. At least 4 other countries in favour of extension cut-off to be the autumn NB EU leaders need to reach unanimity by law— katya adler (@BBCkatyaadler) April 10, 2019
Earlier Mrs May said she "regretted" that the UK had not already left, adding that she had "been clear" with the EU that she was only seeking a short delay to Brexit.
Ahead of the summit, European Council President Donald Tusk said "neither side should be allowed to feel humiliated" and urged the other 27 leaders to back a flexible extension of up to a year.
Mrs May said the UK could leave the EU whenever a deal was ratified by Parliament - meaning the exit date could be by 22 May - the day before the European Parliament elections.
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