A herder saw the group of British citizens at the Ara-ta camp in Eritrea and reported it to the Ethiopians, according to an official, who asked not to be named.
The Foreign Office confirmed on Saturday that a team of British officials had arrived in Addis Ababa, Ethiopia's capital, saying only: "They have arrived. We won't comment on the composition of the team."
The whereabouts of a separate group of up to 10 French citizens, earlier reported kidnapped, were still unclear though the group's Ethiopian tour operator was reported as saying that the French had not, in fact, been kidnapped.
"They are safe. It is the first contact we have with them since last Tuesday," Samson Teshome, the head of tour group Origins Ethiopia, said.
He said the group would probably reach the northern city of Mekele late on Saturday or on Sunday.
"We are very lucky and happy that they are safe," he said.
The missing Europeans were part of two separate tour groups - one from the British embassy and the other of French tourists - who were travelling in the Afar region near the border with Eritrea.
Earlier, Margaret Beckett, the British foreign secretary, said that five "members of staff, or relatives of members of staff, at our embassy in Addis Ababa" were missing.
She made no direct reference to them being kidnapped, but said Ethiopia had promised "to ensure that the situation is resolved peacefully".
An Ethiopian government source told the AFP news agency: "The army is dealing with the search. Troops have been deployed in the region."
Ethiopia's state news agency reported police as saying that the Britons were taken from a camp on Thursday night.
French embassy officials have also travelled to the region in an attempt to help trace the French nationals who were reportedly part of a trekking group.
The missing Ethiopians were working as drivers and interpreters for the tour groups.
The government requires all convoys visiting the area to have a minimum of two cars and travel with armed guards because of the threat of bandits.
"We heard about an abduction and we are trying to confirm their whereabouts," Berekhat Simon, an adviser to Meles Zenawi, the Ethiopian prime minister, told AFP.
"We'll try our best to ensure their safety."