In a statement, the AKP and the far-right Nationalist Action Party (MHP) said: "The issue is a bleeding wound in higher education and needs to be resolved."
 
It said the two parties "have held negotiations to resolve the issue and come to a general agreement. Technical work is continuing".
 
The AKP has 340 seats in the 550-member parliament and the MHP 70.
 
With their forces combined, they would easily be able to garner the two-thirds majority of 376 votes that would be needed to amend the constitution.
 
Legal reform
 
The AKP, the offshoot of a now-banned Islamist party, has long opposed the headscarf ban in universities, arguing that it violates both the freedom of conscience and the right to education.
 
The MHP recently announced that it too favoured a legal reform to end the restriction.
 
Secularist forces, including the army, senior judges and many academics, see the headscarf as a symbol of defiance against Turkey's fiercely guarded secular system.
 
The joint statement did not detail what amendments the two parties have in mind.
 
Equal treatment
 
Newspaper reports said the AKP suggested amending two articles of the constitution to read that everyone will be treated equally in public service and no one will be barred from education because of their attire.
 
Public servants are also banned from wearing the headscarf in Turkey, but newspapers quoted a senior AKP politician as saying the planned reform would only extend to university students.
 
In 2005, the European Court of Human Rights ruled that the headscarf ban in Turkish universities was not a violation of fundamental freedoms and could be necessary to protect Turkey's secular order against extremist movements.