(CNN) -- Sudan and South Sudan may be sliding back toward war, the United States and other international powers are warning, amid reports that Sudan is bombing its newly independent neighbor.

The White House is "alarmed" by recent fighting in the region of Southern Kordofan, Sudan, it said in a statement Tuesday, urging both sides to "exert the greatest restraint."

U.S. Secretary of State Hillary Clinton laid most of the blame for the recent fighting on Sudan.

On Tuesday, she called bombing runs and the use of heavy weaponry by the North "evidence of disproportionate force on the part of the government in Khartoum."

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The United States has also urged South Sudan to stop arming rebel groups in its northern neighbor. South Sudan denies it is doing so.

"We are urging both parties to cease all military activity along the border, because it is a flashpoint that could become even more dangerous and escalate out of control," U.S. State Department spokeswoman Victoria Nuland said Tuesday.

On Wednesday, the top European Union foreign policy official echoed the American concerns.

Catherine Ashton called the clashes "a dangerous escalation of an already tense situation," and warned: "Further cross-border military activity could result in a wider military confrontation."

The ground clashes that erupted this week are the first between the two sides since South Sudan became independent last year.

They have thrown into doubt plans for Sudan's President Omar al-Bashir to visit South Sudan next month for talks with South Sudan President Salva Kiir.

South Sudan military spokesman Philip Aguer Tuesday accused Sudan of bombing an oil field in the south, a claim Sudan army spokesman Al-Suwarmi Khalid denied.

Actor George Clooney, an outspoken activist on Sudan, was arrested Friday protesting outside the country's embassy in Washington.

Clooney met with U.S. President Barack Obama Thursday to discuss his concerns about Sudan after testifying before the Senate Foreign Relations Committee about violence in the Nuba Mountains of South Kordofan.

On Monday, the United Nations Security Council "demand(ed) that all parties cease military operations in the border areas and put an end to the cycle of violence," said Security Council President Mark Lyall Grant of the United Kingdom.

"The members of the Security Council are deeply alarmed by the military clashes in the region bordering Sudan and South Sudan, which threaten to precipitate a resumption of conflict," Lyall Grant said.

South Sudan became independent from Sudan last year after years of civil war over oil-rich territory.

Recent talks between the two sides failed to resolve the long-running dispute over oil revenues.

South Sudan shut down oil production in late January after accusing its northern neighbor of stealing $815 million worth of its oil. Sudan said it confiscated the crude to make up for unpaid fees to use the pipeline and processing facilities in its territory.

Clashes this month included "ground fighting on both sides of the border and aerial bombardment," the African Union said Tuesday.

"Military means will never provide a long-term to answer to the bilateral issues affecting the relations between the two countries," AU Chairperson Jean Ping said Tuesday.

Sudan's ambassador to the U.N. denied his country was bombing South Sudan and insisted that it had a right to defend itself.

"When our security is threatened inside our territories by rebel movements, we have every right to use all possible means to repel and put an end to those attacks which threaten our security," Daffa-Alla Elhag Ali Osman said.

It is not clear how many have been killed in the recent clashes, but the U.N. refugee agency says thousands have fled the violence.

The two sides signed a Memorandum of Understanding on Non-Aggression and Cooperation just last month, and are due to hold a presidential summit starting April 3.