The proposal will lift the ban only on the traditional headscarf tied under a woman's chin.
Bahceli 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."
Teachers and public office workers will still be banned from wearing all types of headcovering.
The move has drawn criticism from judges and university rectors in Turkey's powerful secular elite.
The AK party has wanted to change the law banning the headscarf in universities, first introduced after a military coup in 1980, for many years.
|Turkey teacher's headscarf right|
During his summer 2007 election campaign, Recep Tayyip Erdogan, the prime minister, vowed to bring in changes.
Gaining the nationalist MHP's support has meant they have the backing needed to change the constitution.
It is argued by the secular Republican People's party that relaxing the ban would damage Turkey's secular traditions.
The dispute instigated early elections last year and any raising of tensions between secularists and the government could affect the country's EU candidacy and economic conditions.
The joint decision on changing the constitution and higher education laws was made late on Monday.
A two-thirds majority would be needed in parliament to make a constitutional change - a majority that the two parties have.