Abadi said in Baghdad, a staunch opponent of the Kurdish move, that he would not negotiate on the back of the referendum result -- expected to be a resounding "yes".
But in Kurdish regional capital Arbil, Barzani in a televised address urged the Iraqi premier "not to close the door to dialogue because it is dialogue that will solve problems".
Iraqi Kurds have cast their votes in a historic referendum in what they see as a major step towards a long-cherished dream of statehood but amid widespread international concern, Iraq's prime minister has denounced the move as unconstitutional.
"We assure the international community of our willingness to engage in dialogue with Baghdad," he said.
"The referendum is not to delimit the border (between Kurdistan and Iraq), nor to impose it de facto," Barzani added.
72 hour deadline
Iraqi Prime Minister Haider al-Abadi on Tuesday gave authorities in Kurdistan 72 hours to give the central government control of airports, a day after the autonomous region voted on independence.
Speaking at a news conference, Abadi said his government would ban "international flights to and from Kurdistan" in three days unless the airports are placed under its control.
Two airports operate in the autonomous Kurdish region of northern Iraq, at regional capital Arbil and in second city Sulaimaniyah.
The Kurdish referendum went ahead despite both Iraqi and international opposition.
Iraq's Kurds were Tuesday expecting the announcement of a big "yes" vote for independence, as authorities in Baghdad weighed how to respond to a referendum they considered illegal.
Large numbers turned out in northern Iraq's autonomous Kurdistan region for Monday's vote, which went ahead despite fierce objections from Baghdad, Turkey and Iran.
Votes were still being counted on Tuesday, with results expected by the end of the day and no doubt of an overwhelming outcome in favour of independence.
The vote is non-binding and will not lead automatically to independence, but is seen by the Kurds as a major step towards a long-cherished dream of statehood.
In the regional capital Arbil, a night of fireworks, flag-waving and dancing in the streets followed the vote.
"We made a Kurdish state today," Arbil resident Ahmad told AFP during the celebrations.
"We're Kurdish people, we're not Arab, we're not Persian, we're no one else... We're Kurds and we'll remain Kurds forever."
The referendum took place peacefully, but has increased tensions between the Iraqi Kurds and their neighbours, raising fears of potential unrest.
Iraq Prime Minister Haider al-Abadi declared before the vote that he would take "necessary measures" to protect the country's unity and he was due to address parliament on Wednesday.
Iraqi lawmakers voted on Tuesday to send troops to disputed areas where the referendum took place but there have been no signs of a deployment so far.
Turkey warns of 'ethnic war'
Analysts say Baghdad is deeply concerned by the vote but unlikely to seek a confrontation with the Kurds for now, especially as Iraqi forces continue to battle the Islamic State group in its final bastions.
Turkey, concerned the vote will stoke the separatist ambitions of its own sizeable Kurdish population, repeatedly condemned the vote as wrong-headed and dangerous.
On Tuesday, President Recep Tayyip Erdogan warned the Iraqi Kurds and their longtime leader Massud Barzani risked sparking an "ethnic war".
"If Barzani and the Kurdistan Regional Government do not go back on this mistake as soon as possible, they will go down in history with the shame of having dragged the region into an ethnic and sectarian war," Erdogan said in a televised speech.
Erdogan warned on polling day that Turkey would shut its border with Iraqi Kurdistan and threatened to block key exports that flow from the region through Turkish territory.
He even suggested the possibility of a cross-border incursion similar to the one Turkish forces have carried out against IS and Kurdish fighters in Syria.
The vote took place in across the three northern provinces of autonomous Iraqi Kurdistan -- Arbil, Sulaimaniyah and Dohuk -- as well as in disputed border zones such as the oil-rich province of Kirkuk.
Kirkuk curfew lifted
A curfew was lifted early Tuesday on parts of the city of Kirkuk, where it had been imposed on the city centre and non-Kurdish neighbourhoods due to fears of unrest connected with the vote.
An AFP journalist saw heavy traffic and shops opening as normal in the city of about one million, which is outside the boundaries of Iraqi Kurdistan and divided between Arab, Kurdish and Turkmen populations.
Officials reported that turnout for the referendum stood at 72 percent, with 3.3 million of the 4.58 million registered voters taking part.
Participation was lower in some parts of the region and was at only 50 percent in Sulaimaniyah province, the home base of political forces opposed to Barzani.
Barzani's opponents have accused the longtime regional chief of seeking to empower himself through the vote and said he should have accepted a UN-backed plan to put off the referendum in favour of negotiations with Baghdad.
The United Nations and United States urged Barzani to cancel or postpone the vote, with Washington especially concerned it could hamper the fight against IS in which Kurdish peshmerga forces have been vital.
Largest stateless people
Issam al-Fayli, a political science professor at Baghdad University, said he didn't expect any immediate confrontations.
"The Iraqi government will take its time to make decisions, taking special account of the opinions of military leaders because the current priority for Iraq is the war against Daesh," he said, using an Arabic name for IS.
"There will be some minor incidents but the crisis should in the end remain under control."
Turkey was also likely to remain cautious, said Dana Nawzar Jaf, a researcher on Middle East politics at Durham University in England.
"Turkey right now is not willing to cause a collapse of the KRG. The KRG represents a buffer zone between Turkey and Iran and represented Turkey's strongest ally in Iraq," he said.
Iran, which like Turkey has a large and restless Kurdish population, also opposed the referendum and on Sunday announced it was blocking all flights to and from Iraqi Kurdistan at Baghdad's request.
Left without a state of their own when the borders of the Middle East were redrawn after World War I, the Kurds see themselves as the world's largest stateless people.
The non-Arab ethnic group number between 25 and 35 million people spread across Iraq, Iran, Turkey and Syria.