Witnesses from the northern border town of Razah also said that the Saudis had begun an offensive on Monday.
“The Saudi army launched a vast offensive against Houthi positions in the border region,” one witness, who asked not to be identified, told to the AFP news agency.
Hashem Ahelbarra, Al Jazeera’s correspondent in northern Yemen, said Saudi Arabia’s incursion could be a rescue operation.
“The Houthis said yesterday that they had managed to kill and take some Saudi soldiers prisoner,” he said.
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“The Saudis [are] insisting … that the aim of the air strikes and military campaign is not just to pound Houthi areas but to make sure the Houthis are not going back to the borders to launch attacks against Saudi Arabia.
“Are they going to be able to achieve that? We have to wait and see because the area itself is extremely difficult; it is rugged terrain where the Houthis have been … using guerrilla war [tactics] to wage attacks against the Yemeni government and at the same time Saudi Arabia.”
Prince Nayef bin Abdel Aziz Saudi, Saudi Arabia’s interior minister, said on Monday that “other countries should not interfere” in Yemen’s affairs.
“We condemn any interference in Yemen, and any action is to be taken by Yemenis themselves,” said the minister.
“But when it comes to crossing the borders and violating the security of the Saudi kingdom, we will defend the Saudi territories even if it is just one metre.”
Yemeni officials also denied that Saudi troops had begun an assault inside their country.
“Each side is dealing with the rebels on its own territory,” Colonel Askar Zaayl, a Yemeni army spokesman, said.
“It’s not true that the Saudis are launching strikes and attacks inside Yemeni territory.”
It was not immediately clear whether either side had suffered any casaulties in the latest fighting, but the Houthi rebels claimed “the aggressors suffered heavy losses”.
Saudi Arabia began bombarding suspected Houthi positions earlier this month after they apparently crossed the border and seized control of a small area.
The Houthis say that the Saudis have been allowing Yemeni troops to use the area to attack their positions.
Saudi officials claim that the fighting in northern Yemen is being supported by Iran and could be helping al-Qaeda to cross to their side of the border.
“It cannot be ruled out that there are contacts or co-ordination between them,” Prince Nayef was quoted as saying by the Asharq al-Awsat newspaper.
The Shia Muslim Houthi fighters, citing political, economic and religious marginalisation, have been battling the government of Ali Abdullah Saleh, the Yemeni president, since 2004.