"We are particularly concerned about the worsening situation in West Darfur," Raouf Mazou, the U.N. refugee agency's (UNHCR) assistant high commissioner for operations, told a briefing in Geneva.
"According to reports from colleagues on the ground, the conflict has reached alarming levels, making it virtually impossible to deliver life-saving aid to the affected populations," he stated.
Mazou also said "Increasing numbers of injured civilians are among the newly arrived refugees in Chad," including women and children.
He added: "The ethnic dimension that we had observed in the past is unfortunately coming back" in Darfur, a region scarred by civil war in the 2000s.
Darfur, a vast western region bordering Chad, has witnessed the deadliest violence since the conflict between rival armed forces broke out on April 15.
The Sudan army, led by Abdel Fattah al-Burhan, has been battling the paramilitary Rapid Support Forces (RSF) commanded by his former deputy, Mohamed Hamdan Dagalo, after the two fell out in a power struggle.
The Darfur conflict, which erupted in 2003, pitted ethnic minority rebels who complained of discrimination against the Arab-dominated government.
"What we are seeing today is a widening and intensification of conflicts," said UNHCR's Mazou.
"It is very worrying and we are calling on all parties to the conflict to ensure that they spare civilians."
UNHCR said the number of refugees fleeing the fighting in Sudan had now topped 560,000, while the number of people internally displaced within the country was around two million.
UNHCR said it was increasingly alarmed by the growing humanitarian needs of people affected by the crisis, as the delivery of aid remains severely limited by insecurity, a lack of access and low funding.
"560,000 people in just over two months is a huge number," said Mazou.
He said that level of burden could frighten some neighbouring countries, so the international community had to ensure those states could meet the humanitarian needs.
Egypt has received the highest number of refugees, followed by Chad, South Sudan, Ethiopia and the Central African Republic.
UNHCR was planning on the basis that a million refugees might flee Sudan in six months, but Mazou said given current trends and the situation in Darfur, it is likely the figure will be exceeded.
It had thought 100,000 might reach Chad in six months, but the projection is now 245,000.