Turkey has warned the UN it may need help if the flow of refugees from Syria continues at its current rate.
After speaking to the UN secretary general, Foreign Minister Ahmet Davutoglu said the refugee issue was becoming an "international problem".
Some 2,800 Syrians have crossed in 36 hours, with the total now near 24,000.
Troops and rebels were engaged in heavy fighting across Syria on Friday, just six days before the UN hopes a full ceasefire will come into effect.
The UN estimates more than 9,000 people have died in the year-long uprising against the rule of President Bashar al-Assad.
Rate 'doubles'Mr Davutoglu said he had spoken to UN Secretary General Ban Ki-moon early on Friday to express concern about the rate of refugees.
In later televised remarks, Mr Davutoglu said: "We have spared no efforts to accommodate Syrians fleeing the violence back home, but if they continue to arrive at this rate, we will need the UN and international community to step in."
The UN and Arab League envoy Kofi Annan has set out a six-point peace plan that Syria has accepted, part of which envisages a UN-supervised cessation of armed violence by all parties from 10 April, with a full ceasefire on 12 April.
However, the US and many regional countries have expressed scepticism that Syria is committed to the plan.
Mr Davutoglu said the rate of refugees had doubled since Mr Assad agreed to implementing Mr Annan's plan.
"Those who are giving [Mr Assad] time should also know that the number of fleeing Syrians is increasing and this is turning into an international problem," Mr Davutoglu said.
There appears to be no halt in the violence in Syria.
On Friday, the Syrian Observatory for Human Rights and Local Coordination Committees groups monitored reports from many parts of the country, including:
Shelling, including tank fire, in Douma, near Damascus. Shelling in Zabadani, north-west of the capital
Rebel fighters confront tanks trying to enter Rastan, north of Homs
Protests after Friday prayers in the eastern province of Hasakeh, in the town of Qamishli in the northern Kurdish region and in the eastern city of Deir Ezzor. Snipers on roofs in Damascus and Hama
Heavy fighting in central Homs province - in the villages of al-Tiba, al-Qabu and Shniyeh
Fighting between soldiers and army deserters in Dmeir, 40km (25 miles) north of Damascus
International media cannot report freely in Syria and it is impossible to verify reports of violence.
The Syrian government says 2,000 security personnel have been killed in the uprising and blames the violence on "armed gangs" and "terrorists".
The BBC's Jim Muir in Beirut says that in Homs - much of which is a real battlefield - and some other areas, especially Idlib province, the approach of the deadline to halt all violence seems to have brought an escalation rather than a reduction.
He says activists are accusing the government of trying to complete its crackdown come what may; but government officials say it is the rebel fighters who are exploiting the impending withdrawal of the military from towns and cities.
Syria's official Sana news agency said on Friday that Damascus had sent a message to the UN saying that "terrorist acts committed by the armed terrorist groups in Syria have increased during the last few days" and demanding a written commitment that the opposition would not try to make territorial gains.
A UN team is currently in Damascus to negotiate the possibility of deploying UN monitors to oversee any ceasefire.
Mr Annan has said that if the truce is successful a small mobile UN monitoring mission of some 200-250 observers could be brought into Syria.
Mr Annan is also expected to travel to Iran on 11 April, the day after the partial ceasefire is due, to try to win further regional support for his peace plan.