With 98.2 percent of the ballot boxes opened, 51.31 percent of the votes were for "Yes", backing the referendum, the state-run Anadolu agency reported.
If approved in the referendum, the changes would transfer executive powers now held by the prime minister to the president.
The referendum has split Turkish society, according to voters days before the vote. A victory would echo the forces that powered Donald Trump to the White House, pushed Britain out of the European Union and put Marine Le Pen within shouting distance of the French presidency. The post of Prime Minister would be abolished after the 2019 national elections, and time-limits on the newly-empowered presidency would be reset.
Turkey's President Recep Tayyip Erdogan, center, gestures with the members of his party before addressing his supporters during a referendum rally in Erzurum on April 12. "Why one person? What if he makes a mistake, what if he is deceived, what if he is bought?" said Kemal Kilicdaroglu, head of the main opposition Republican People's Party, during a "no" rally in Ankara Saturday.
'God willing, the result will be auspicious and we will all have the chance to discuss Turkey's fundamental problems'.
Observers from the 57-nation Organization for Security and Co-operation in Europe (OSCE) are monitoring the voting process in the Turkish capital. "Every vote you cast tomorrow will be a cornerstone of our revival", he told a crowd of flag-waving supporters.
Western countries have criticized that tough response, and relations with the EU - which Turkey has been negotiating to join for a decade - hit a low during the campaign when Erdogan accused European leaders of acting like Nazis for banning referendum rallies in their countries on security grounds.
But in the seventh referendum in its history today, the country is more markedly divided and the spoils are much greater.
"However, this referendum is a decision of a new administrative system, a change and a transformation in the Republic of Turkey".
Erdogan has said a new constitution would ensure that the Turkish political system would remain stable and secure, and be similar to the presidential systems of the "world's strongest nations".
Baydemir earlier in the week told Rudaw Turkey's policy against the Kurds has not changed for the past one hundred years, even though different political parties, both secular and Islamic, have assumed power in different eras.
More than 55.3 million Turks were eligible to cast ballots on sweeping changes to the president's role which, if approved, would grant Erdogan more power than any leader since modern Turkey's founder Mustafa Kemal Ataturk and his successor Ismet Inonu.
In the referendum, a "yes" vote will validate Erdogan's long-standing ambitions, and the AKP's bureaucratic monopoly but a "no" vote will not reverse what is already in place. In the predominantly Kurdish province of Diyarbakir, voter Mehmet Sayar didn't say which way he voted, but added: "I hope the result will be the best for our country because this referendum will determine the future of our children". His wife Zeynep agreed, saying: "I was going to come sleep here last night to vote at first light".
A victory for the "Yes" vote will give the president new powers to assign ministers, high-level state officials and vice-presidents, as well as half the members in the country's highest judicial body.
He described the new system as "a bus with no brakes and whose destination is unknown".
Opponents fear it risks granting Erdogan authoritarian powers and allow him to ride roughshod over key institutions like the judiciary and parliament. The Turkish government says the proposed constitutional amendments will bring strong leadership amid the existing terrorist threat and will prevent a return of the fragile coalition governments of the past, which have often paralyzed the executive branch.
Polls in the east opened at 7 a.m. (0400 GMT), while those in the west are to open an hour later.