The new constitution, which will replace the basic law drawn up in the wake of Turkey's 1980 military coup, seeks to set up for the first time a formal presidential system for ruling the country.
Critics have claimed that the move is part of a power grab by Erdogan for one man rule in the wake of the July 15 failed putsch bid.
But Erdogan and the ruling Justice and Development Party (AKP) say the system would bring Turkey into line with countries such as France and the United States and is needed for efficient government.
The two readings are expected to last a total of 13-15 days, it added.
The AKP needs more than 330 votes -- a three fifths majority -- for the bill to then be submitted to a referendum for public approval.
However November 2015 polls left the AKP short of a super majority in parliament and is relying on support of the opposition right-wing Nationalist Movement Party (MHP), the fourth largest party.
Under the new constitution, the president would not have to sever links with a political party, as is the case now even though Erdogan co-founded the AKP.
It is also expected to lead to the creation of posts of vice presidents and the abolition of the office of prime minister.
The number of MPs would be increased to 600 from 550 and the age of eligibility to become a lawmaker reduced to 18 from 25.
The draft law says the president would be elected for a five-year term and serve for a maximum of two mandates.
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