An EU official said most are thought to have been trekking in the Everest or remote Langtang regions. Many are hoped to be alive but isolated by the quake.
The fate of thousands of Nepalese in remote communities is also unknown.
Nepal has called for more foreign help and humanitarian aid, admitting it was ill-prepared for the disaster.
Information Minister Minendra Rijal told the BBC that “seminars and workshops probably are not that useful” in dealing with the crisis.
“And any projection has not taken into account… that so many people and so many districts would be affected,” he said. “This is a calamity of enormous proportions.”
At Kathmandu’s historic Durbar Square, soldiers and volunteers form human chains to remove the debris, brick by brick. The bricks come from historic buildings levelled by the earthquake. Many are very old and are being stored so that they can be used to rebuild these ancient sites.
The soldiers are joined by aid workers – but also tourists. One French visitor said she “just wanted to help”. But it’s an ad-hoc approach which characterises the entire relief operation.
I met rescue and medical teams from France and China. After wandering around they left. “We don’t know what we are supposed to be doing,” one French rescue worker told me.
The Nepalese authorities say the death toll from Saturday’s 7.8-magnitude quake could rise to 10,000.
Nearly 14,000 people were injured in the disaster. Relief and rescue teams have reached most areas but many people remain in urgent need of humanitarian assistance.
The quake struck during peak trekking season in a country popular with mountaineers. Twelve EU citizens are known to have died so far. Some of them were killed by avalanches, triggered by the quake, that struck base camp below Mount Everest.
The EU envoy to Nepal, Rensje Teerink, said the authorities did not know the whereabouts of some 1,000 other EU citizens.
“They are missing but we don’t know what their status is,” she told reporters in the capital, Kathmandu. Another EU official, speaking to the AFP news agency on condition of anonymity, said most of the missing were likely to be found safe.
Many backpackers do not register with their embassies in Nepal, which has made it harder to trace them, Reuters news agency reports.
Landslides and poor weather have hampered efforts to deliver aid to isolated districts.
“Almost every house in my village is destroyed, and 20 people died,” Kumar Ghorasainee told the AFP news agency amid the ruins of his hometown of Melamchi in Sindhupalchok, north-east of Kathmandu.
The region has been identified by the Red Cross as one of those worst-affected.
“We lost our cattle and our sheep,” the 33-year-old English teacher said. “No-one has come to help us – the cars and the aid trucks just drive by.”
There are only about 20 helicopters available for the rescue and relief operations taking place in a region that stretches hundreds of kilometres.
Fights have broken out with victims demanding evacuation being left unable to board helicopters, the BBC’s Sanjoy Majumder reports.
China is expected to send more aircraft, but still more will be needed – along with road vehicles to carry relief supplies.
The UN on Wednesday launched a “flash appeal” for $415m (£270m) to assist Nepal over the next three months – but on Friday it told the BBC it had so far received only $5.8m – 1.4% – in confirmed funding.
There has been growing anger at the government’s response to the disaster, with a number of protests breaking out.
The tent cities which sprang up around Kathmandu in the days following the quake have almost gone, reports say, as the fear of aftershocks subsides.
But the stench of rotting bodies in the rubble has prompted an official order for unclaimed bodies to be cremated immediately, Reuters news agency reported.
The UN has also warned of the challenges facing Nepal’s farming community, which comprises up to two-thirds of the country’s 27 million people.
It says that the quake destroyed seed stocks for the mid-May rice-sowing season, as well as stone huts that were storing grain.
If farmers miss this month’s planting season, they will be unable to harvest rice – Nepal’s staple food – until late 2016, the UN says.