The increase in tension occurred on the sixth day of ethnic Serb protests against the ethnic Albanian-led government's decision to require drivers with Serbian registration plates to put on temporary ones when entering Kosovo.
A registration office was torched in the small town of Zubin Potok and another damaged in Zvecan, though there were no casualties, the prime minister said.
An AFP correspondent reported seeing Serb fighter jets fly twice over the border area around noon.
Serb army helicopters were seen Friday flying several times over border posts blocked by ethnic Serb protesters.
And helicopters serving the NATO peackeeping force KFOR have also been making regular flights over the area since the dispute erupted.
"Individuals and groups, whose activities endanger the rule of law and public order, are attacking our state and disturb our peace," Kurti said on his Facebook account.
"Serbia is encouraging them and supports them clearly," the premier said.
"Serbia abuses the citizens of Kosovo in order to provoke a serious international conflict," Kurti charged.
Kosovo declared independence from Serbia in 2008, a decade after the war between ethnic Albanian guerrillas and Serbian forces which claimed 13,000 lives, mostly ethnic Albanians.
Independence has always been disputed by Belgrade, which encourages minority Serbs living in northern Kosovo not to recognise the authority of the government in Pristina.
The Kosovo government on Monday deployed special police forces near the two border crossings with Serbia in order to enforce what it calls a reciprocal measure.
The government justified the ban on Serb plates over Serbia's refusal to allow cars with Republic of Kosovo plates to enter the country.
Belgrade's position is that the decision to mandate temporary plates implies its status as an independent nation.
But hundreds of Serbs demanding the measure be rescinded have used trucks and other vehicles to block traffic toward the border crossings.
The United States and the European Union have called for a de-escalation of tension and for the two parties to return to normalisation talks, which the EU has mediated for about a decade.
The Serbian president said the normalisation process can only resume if Kosovo withdraws the special police forces from the north.
