The bodies of the climbers were discovered on Thursday on the western flank of the ice-covered peak by a search and rescue team from Lauterbrunnen.
The search was begun after the climbers were reported missing on Thursday morning, cantonal police said.
The bodies were recovered with the aid of an Air Glacier helicopter.
The Eiger, 3,970 metres high, is celebrated for the challenges it poses to climbers. At least 65 have died since 1935 trying to ascend its steep north face, which has earned a nickname in German as the "murder wall".
An inquiry into the exact circumstances of the deaths and the identities of the victims is ongoing, cantonal police said.
Because the identity of the climbers was not immediately known their families could not be notified, police added.
The incident brings to eight the death toll of foreign climbers in the Swiss Alps over the past week.
On Tuesday, five German climbers who plunged several hundred metres from near the summit of the 4,010-metre-high Lagginhorn, an Alpine mountain in the canton of Valais.
According to the initial findings, Valais police said the victims, including two children aged 14 and 17, were not roped up when the accident occurred.
Other victims included a 44-year-old man, the father of the 17-year-old, and two young adults, aged 20 and 21.
The victims were part of a group of six climbers.
The other member of the group, a man whose 14- and 20-year-old sons died in the fall, decided to stop 100 metres from the summit because he was not feeling well, police said.
Another foreign climber, 32, died on Sunday morning after a fall on the Wetterhorn, another mountain in the Bernese Oberland, near Grindelwald.
Story continues below…