UK officials say progress is being made "behind the scenes" in the search for five Britons kidnapped four days ago in the Afar region of northern Ethiopia.
Eritrea denies claims its forces took them over the disputed border.
Neither the UK Foreign Office nor the Ethiopian ambassador in the UK have commented on who may be responsible.
The embassy staff and their relatives were sightseeing along with 13 Ethiopian guides near the border when they all went missing.
The state-run Ethiopian News Agency is reporting that five of the Ethiopians - working as drivers and translators - were found near the Eritrean border late on Saturday.
It was not clear whether they had escaped or were released.
It is thought the tourist group stopped near the remote town of Berahle shortly before they were abducted about 37 miles (60km) from there, said the BBC's Amber Henshaw.
Speaking from the town our correspondent said officials were saying very little, but did confirm "some progress" was being made.
Foreign Office Minister Geoff Hoon told ITV1's The Sunday Edition that the department was "working as hard as we possibly can" to resolve the situation.
"It is a matter of grave importance... as it would be with any British tourist," he said.
"But obviously staff in the Foreign Office feel particularly strongly because it is their people."
Head of the Afar region Ismael Ali Sero said cars used by the sightseers were set on fire in the early-morning raid on their camp, about 800km (500 miles) north-east of Addis Ababa.
He said about 25 Eritreans in military uniform marched the group 12-18 miles (20-30 km) to Waime in Eritrea, and a local herder reportedly saw them at the Ara-ta military camp in Eritrea.
But Berharnu Kebede, Ethiopian ambassador to the UK, said: "We are not in the business of finger pointing at any group or individual or any country.
"For us the priority is to secure the safe return of these people."
Yemane Gebremeskel, of Eritrean President Issaias Afeworki's office, said Mr Ismael's claim was "crazy".
"No one is involved in any business of kidnapping," he told the AFP news agency.
The area is one of hottest places on Earth (picture: Sam Vader)
A peace deal was struck in 2000 but the two countries have yet to define the status of the border, patrolled by UN observers.
The BBC World Service's Africa Editor, David Bamford, said the claims of Eritrean involvement were being met with scepticism.
"It doesn't seem to be in the Eritrean government's interest at all to be engaged in anything like this."
He said the allegations were coming from local party officials in Ethiopia, who could be seen as having an "axe to grind".
If Eritrean soldiers were involved it could only be explained as a "mistake", he said.
It was more likely a local rebel group had abducted the Britons, but he said local internal politics and the remoteness of the region would make it difficult for British officials to establish what was going on.
Tourists visit the area mainly to see the Danakil Depression, one of the lowest and hottest places on Earth known for its salt mines and active volcanoes.
But bandits and rebel groups operate there and travellers are advised to travel with an armed guard.