Gabriel Yei, director of Caritas South Sudan, said that Dominic Pathic, James Celestino and William Atorjony when security forces confronted them and took them into custody. No word on their whereabouts was available.
A Catholic source in Khartoum confirmed the arrests, the network reported.
The arrests came as tensions eased slightly along the tenuous Sudan-South Sudan border near the disputed town of Heglig -- known as Panthou in South Sudan -- and nearby oil fields. South Sudanese leaders withdrew from the town April 20 at the behest of the United Nations and the African Union.
Sudan's leader Omar al-Bashir later said his forces had retaken Heglig.
The escalating fighting and rhetoric between the two countries had led to fears of all-out war.
South Sudan had claimed the area April 9 under a 1956 agreement that the South believes set the border north of the town. Heglig had been used as a staging ground for military assaults by Sudan on South Sudan, particularly in the nearby city of Abyei, since May 2011.
South Sudan President Salva Kiir warned, however, that his country's forces "will hit back hard" should cross-border attacks by Sudan resume.
In Khartoum, a crowd of angry Muslims attacked a Protestant church compound used by southerners April 21 after a Muslim preacher known for fiery sermons took advantage of the excited climate to call for "jihad" against Christians during Friday evening prayers, Youssef Matar, secretary general of the Presbyterian Evangelical Church told Reuters.
The crowd reportedly ransacked the contents of a church in the compound and set it afire, witnesses said. No injuries were reported.
Hundreds of thousands of southerners and people from regions along the disputed border remain in Khartoum, Sudan's capital, many of them in a legal state of limbo since South Sudan became an independent nation in July.