The bombing took place on the Yemeni side of the border close to the Saudi city of Najran, they said. Clashes were also seen in the northwestern Yemeni town of Haradh which borders the kingdom, witnesses told Reuters.
Yemen's Houthi-run state news agency, Saba, said Houthi forces had fired missiles at Saudi targets.
The flare-up in fighting was one of the worst since peace talks began in Kuwait in April between Yemen's government and the Houthis to end a 16-month conflict that has left more than 6,400 people dead, nearly half of them civilians, and displaced more than 2.5 million.
A truce that began in April has slowed the momentum of fighting, but violence continues almost daily.
Prospects for progress in the talks dimmed further on Thursday when Houthi rebels and their allies in the General People's Congress (GPC), the political party of militarily powerful former president, Ali Abdullah Saleh, said they had decided to form a political council to unilaterally rule the country.
Ismail Ould Cheikh Ahmed, the United Nations Yemen envoy, said the move gravely violated UN Security Council Resolution 2216, which calls on the Houthis ‘to refrain from further unilateral actions that could undermine the political transition in Yemen’.
On Friday, the delegation of the internationally recognised government of President Abd-Rabbu Mansour Hadi said it planned to pull out of the talks on Saturday. The decision was taken apparently in protest at the announcement of the formation of the council.
But on Saturday afternoon, Cheikh Ahmed said on his Twitter account that he had met both delegations, and ‘suggested a one-week extension to the talks and a framework for a solution to the crisis in Yemen’.
The official Kuwait news agency reported that both delegations were studying his proposals.
Kuwait on July 20 set a 15-day deadline for its hosting of the current round of UN-brokered negotiations, which began in June, saying if the parties could not reach a peace agreement by then the Gulf state would have to be ‘excused’ from its role.
A statement by a group of ambassadors accredited to Yemen was posted on the website of the US embassy describing the announcement of the Houthi-GPC council as being inconsistent with the ‘good faith pursuit’ of peace. The ambassadors, who were not identified, said the negotiations represented the best hope for bringing peace to Yemen.