"We take any and all attempts against the National Security of the United States seriously, and will continue to investigate any additional personnel who may be manipulating their privileges of residence", she added.
Ortagus also said that no members of the Cuban mission are permitted to travel outside of Manhattan, where the United Nations headquarters is located.
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Cuba's Foreign Minister Bruno Rodriguez on Thursday slammed the expulsion by Washington of two members of its United Nations mission as "unjustified".
Under the Obama administration, however, Cuban-U.S. relations began to normalize but have since worsened under President Donald Trump who accuses Cuba of propping up embattled Venezuelan President Nicolas Maduro.
Mr Rodriguez tweeted (in Spanish) that the accusations were "vulgar slander", and said the expulsion has created tensions between the two countries.
A view of the U.S. and Cuban flags prior to the signing of agreements between the Port of Cleveland and the Cuban Maritime authorities in Havana, Cuba, October 6, 2017.
"Given that the situation is ongoing, we will not comment further at this stage other than to confirm that we will closely follow this matter", the United Nations said in a statement.
Over the decades, Washington and Havana, old Cold War foes, have engaged in tit-for-tat expulsions of diplomats.
The mysterious symptoms suffered since 2017 caused a diplomatic crisis between Cuba and the United States after Washington that year pulled non-essential embassy staff from the Cuban capital.
The administration announced a new ban on United States tour groups travelling to the Caribbean island nation in June.
But relations have cooled off during the Trump presidency.