UPDATE 1240 GMT: A United Nations spokesman said four UN helicopters were fired upon, probably by militias allied to Republic of Sudan forces, during a visit to Abyei late on Tuesday.
Hua Jiang said 14 rounds were fired when the helicopters took off, but the crews landed safely.
Jiang said militias of the Arab Misseriya tribe were probably responsible for the attack, adding that they were now moving southwards after civilians had left the main settlement of Abyei.
UPDATE 0745 GMT: On Tuesday, President Bashir said that Abyei belongs to the Republic of Sudan: "Abyei is northern Sudanese land. We will not withdraw from it."
Bashir said he had authorised his army to respond to any possible "provocation" by the army of South Sudan.