The workers had distributed aid the insurgents allege is linked to Saudi Arabia.
In a statement, the NRC said authorities in the rebel-controlled Red Sea district of Hodeida detained three Yemeni staff and a contracted driver on Tuesday last week.
"Due to the security sensitivities regarding our staff, we cannot comment any further on the matter at this time," said the Oslo-based non-governmental organisation.
In a statement carried by Yemen's pro-government sabanew.net news website, Local Affairs Minister Abdul Raqib Fattah had said earlier a dozen staff had been "abducted" from the aid group's offices in the Hali district of Hodeida last week.
Local sources told AFP the employees were accused of having accepted and distributed aid from a Saudi-led coalition, which has been battling the Huthis since March 2015.
The NRC said it did not take funding from Saudi Arabia in any of its operations but had been using recycled boxes to distribute hygiene kits in Yemen.
The group said when staff had opened the boxes to distribute the aid, the inside read "The campaign of the Custodian of the Two Holy Mosques for emergency response," a reference to the Saudi king.
The boxes were dated January 2015 and had originally been used for food, according to the NRC.
Yemen's conflict pits a Saudi-led Arab coalition supportive of President Abedrabbo Mansour Hadi against Iran-backed Huthis, who currently control the port of Hodeida along with the capital Sanaa and large parts of northern Yemen.
Monday's news comes amid a push by forces loyal to Hadi, backed by the Arab coalition, to close in on Hodeida, located on Yemen's western coast.
The loyalist forces took full control of Mokha, south of Hodeida, earlier in February as part of a major offensive to oust the Huthis and their allies from Yemen's southwestern coast.
The conflict escalated in March 2015 when the Saudi-led coalition began air strikes to help pro-Hadi forces to take large parts of the country back from the rebels.
More than 7,400 people have been killed and nearly 40,000 wounded in two years of fighting in Yemen, according to the World Health Organisation.
UN humanitarian coordinator for Yemen James McGoldrick in January said more than 10,000 civilians had been killed since 2015.