The former Turkish prime minister, Bulent Ecevit, has been asked by President Demirel to form a new government.
The minority coalition led by Mesut Yilmaz was defeated in a confidence vote last week.
After the meeting Mr Ecevit said he considered this a great honour and that he would accept the challenge.
The BBC correspondent in Ankara says Mr Ecevit is expected to form a coalition with the two main conservative parties.
The new government would then take Turkey through to early elections in April.
From the newsroom of the BBC World Service