At least 4,300 people have now fled violence in Syria to seek refuge in Turkey, a senior Turkish official says.
A BBC correspondent on the border says the real number of displaced people is probably much higher.
The Syrians have mainly been fleeing the town of Jisr al-Shughour, targeted in a government crackdown.
The US said the "Syrian government's offensive in northern Syria has created a humanitarian crisis".
A White House statement call for the violence to stop, and the International Committee for the Red Cross to be given "immediate, unfettered access to this region".
An eyewitness described a tank attack on a village near Jisr al-Shughour on Friday morning, in which people were killed and crops destroyed.
Syria's government says its forces went into the town to restore order after the deaths of 120 security personnel.
Clashes throughout the country on Friday led to the deaths of at least 32 people.
Hundreds of people have been killed in a crackdown in recent weeks on anti-government protests, which began in March.
The BBC's Owen Bennett Jones, in Guvecci on the Turkish-Syrian border, says the real number of displaced people is probably much higher than official total because many have slipped across the border unnoticed by the Turkish army.
Senior Foreign Ministry official Halit Cevik said Turkey would deal with the crisis as best it could.
"If they are coming, this is a humanitarian issue," he said. "We will do whatever is needed within our means."
He added that while Turkey could cope with the crisis at the moment, it might need international help if things got worse.
But Mr Cevik said that the UN refugee agency, the UNHCR, would not be involved.
The Syrians involved were not seeking refuge in Turkey, he said, as their eventual aim was to go back home.
On the outskirts
Some of those arriving at temporary camps across the border inside Turkey have serious gunshot injuries, including a Syrian Red Crescent worker who said he was shot in the back as he tried to help the injured in Jisr al-Shughour.
Both state media and activists on the ground have reported troops and tanks advancing on the town. Most residents are believed to have abandoned it.
The government blamed "armed groups" for the deaths of 120 security personnel in Jisr al-Shughour earlier this week, but some reports said the troops were shot after a mutiny.
Syrian TV said troops had reached the outskirts of the town after securing nearby villages, and that they had killed or captured a number of armed men.
Activists said they had blasted the town with tank fire, but it is unclear how much resistance the troops are facing in an area whose population has largely fled.
Witnesses said troops had been bulldozing homes and torching crops and fields.
Our correspondent says that with all this going on it is inevitable that the exodus into Turkey will continue.
Syria has prevented foreign journalists, including those from the BBC, from entering the country, making it difficult to independently verify reports from there.
Turkey has also so far denied journalists access to displaced Syrians, but Mr Cevik said that when "all is in order" access would be given.