The move has worried secularists, who fear the government is raising the profile of Islam in this Muslim but secular country.
Deniz Baykal, leader of the pro-secular Republican People's Party, called the attempt to lift the ban a "threat against the republic."
The headscarf is a highly sensitive issue in Turkey which is secular but predominantly Muslim.
Much of the country's establishment, including the army, views it as a symbol of political Islam and a threat to the separation of state and religion.
During his summer 2007 election campaign, Recep Tayyip Erdogan, the prime minister, had vowed to bring in the change.
Referring to the bill, which was presented to parliament on Tuesday, Erdogan said the parties had taken "every kind of concern'' into consideration.
He said: "We are taking an important step. Our only goal is to end unjust treatment of girls at the university gates and nothing else."
The AK and MHP delegates had sought the support of secularist parties in parliament during Tuesday's debate.
But Hakki Suha Okay, a prominent member of the Republican People's Party, said the party's opposition to any attempt to lift the ban remained.
Tayfun Icli, from the Democratic Left Party, the other staunchly secular party in parliament which has 13 seats, said: "Those who have open or secret aims against the secular democratic republic, those who want to change the regime are aiming to damage the constitutional system."
Defence of bill
Erdogan rejected the criticism and said: "The government is the guarantor and protector of the republic, secularism, the democracy and the state of law.
"None of our steps or practices have been contrary to that and nor will they ever be."
Devlet Bahceli, leader of the MHP, said "chadors, veils and burqas will not be allowed", referring to Islamic clothing that covers the whole body.
"No one will be allowed to use head scarves as political statements against the state," he said.
"Solving the headscarf issue would relax a large segment of the society. It is a question of rights and freedoms."
Teachers and public office workers will still be banned from wearing all types of headcovering.