Yemen's Shi'ite rebels said they had captured some Saudi soldiers yesterday, after Riyadh said it would press on with its offensive until it had cleared them from its territory.
A Saudi official said on Thursday that Riyadh had launched air strikes on rebels in northern Yemen after the Shi'ite insurgents made a cross-border raid earlier in the week.
But the Saudi Press Agency (SPA) said yesterday the strikes were "focused on infiltrators in Jabal Dukhan and other targets within the range of operations within Saudi territory".
Rebel Yemeni spokesman Mohammed Abdel-Salam told Al Jazeera TV by telephone "We will carry out interviews with them (for the media)... they will be treated with respect," and reiterated his group's position that Saudi forces had crossed the border into Yemen.
Saudi officials were not immediately available for comment. Mr Abdel-Salam did not say how many soldiers were in rebel hands.
Saudi Arabia, the world's top oil exporter, has become increasingly anxious about instability in Yemen, which is facing a Shi'ite insurgency in the north, separatist sentiment in the south and a growing threat from resurgent al Qaeda fighters.
"The entry of the gunmen to Saudi territory, the aggression against border patrols...and presence on Saudi soil is a violation of sovereignty that gives the kingdom every right to take all measures to end this illegitimate presence," SPA reported, citing an official source.
"The operations will continue until all sites within Saudi territory are cleansed of any hostile element."
Riyadh will take unspecified measures to prevent any future incursion by the Yemeni rebels, the source said, adding that armed forces units had been deployed to back border guards.
The rebels accused Saudi Arabia of attacking villages inside Yemen.
"Saudi aircraft are bombing the Malaheet province and the surrounding Yemeni border villages," they said in a statement.
On Thursday, Saudi government officials said the air force had bombed Yemeni rebels who had seized a border area inside the kingdom, which they said had been recaptured. They said at least 40 rebels had been killed in the fighting.
Yemen's government - which has long dismissed accusations by rebels that it has colluded with Riyadh to combat them - has denied that Saudi planes had struck across the border.
At least 40 rebels surrendered during the air strike in the Jebel Dukhan region, an Al Arabiya correspondent told the Saudi-owned network from the region, citing unnamed sources.