General Beshir, speaking on his return from the Organisation of African Unity summit in Ethiopia, said Egyptian police had attacked other police posts in the disputed Halaib Triangle.
The clashes came after Cairo hinted that the Sudanese government may have helped gunmen who tried to assassinate the Egyptian President, Hosni Mubarak, in Addis Ababa on Monday. He was unhurt.
A Sudanese interior ministry statement said seven Sudanese policemen were wounded in the clash on Tuesday. Egyptian security sources had earlier said three Egyptians and three Sudanese were hurt in a clash.
General Beshir condemned Mr Mubarak as "irresponsible" for suggesting Sudan was behind the attempted killing, accusing him of trying to blame others for Egypt's security problems.
In Cairo, Mr Mubarak ruled out sending troops to overthrow Sudan's hardline Islamic government. After meeting exiled Sudanese opposition leaders, he denied plotting with Britain and the United States to overthrow General Beshir's government.
The Sudanese leader said Mr Mubarak was provoking Khartoum by holding the meetings with exiles, including the former president, Jaafar Nimeiri.
Mr Mubarak said: "We are not preparing a coup d'etat and I say to General Beshir: 'We are not intervening in your internal affairs and if we wanted to overthrow your regime, we could do it in 10 days'. Egypt does not intervene in the internal affairs of a country ... but I hope the Sudanese people fulfil their wishes because they are capable of dealing with their leaders, taking the necessary decisions and putting them into practice.
"We are not being asked to intervene militarily because that would lead to deaths among the Sudanese, something we don't want because we think of Sudanese like Egyptians."Reuse content