The cantons, in south-western Switzerland, announced the move at a press conference in Lausanne on Thursday under the banner ‘The Swiss Made Winter Games 2026'.
As yet no host city within the two cantons has been chosen as the focus of the candidacy, though both Lausanne, in Vaud, and Sion, in Valais, have separately expressed interest, said news agency ATS.
The cantons would certainly have plenty of spots to choose as locations for winter sports.
Among the famous resorts in Valais are Zermatt, Verbier and Crans-Montana, while Vaud has Villars, Les Diablerets and Leysin.
However the two cantons are not the only Swiss regions interested in hosting the 2026 games.
The Swiss capital Bern has also thrown its hat in the ring, as will the canton of Graubünden, which includes the ski resorts of St Moritz and Davos- Kloster.
Also touted is the idea of spreading the games out over multiple sites within the country.
The potential candidates now have until December to work on their proposals and submit them to the Swiss Olympic association.
The association will then select which city or region it will put forward to the International Olympic Committee (IOC) as Switzerland's official bid, by September 2017 at the latest, it said in a March statement.
However the decision will be dependent on the outcome of a referendum on the subject in the cantons concerned.
In 2013 the people of Graubünden rejected a proposal to host the 2022 winter games, while in 1988 the residents of Lausanne quashed the city's 1994 candidacy bid in a referendum.
In March this year sports federations in Switzerland gave their backing to Swiss Olympic's proposal that the country should bid for the 2026 games.
Speaking at the time, president of Swiss Olympic, Jorg Schild, said: “With our mountains and our long tradition of winter sports we are destined to welcome the winter Olympic games once again.”
“The Olympic Games could also revitalise tourism in our country,” he added.
Winter sports are a vital tourism magnet for the country, including Valais.
A report published by the Valais Tourism Observatory earlier this week found that hotels and ski lifts were hugely dependent on tourism, with 91.9 of hotel guests and 89.9 percent of ski lift users being tourists.
Cultural institutions, the property industry and restaurants get around half their business from tourists, it said.
However it also found that tourism's contribution to the canton's economy had reduced by around two percent since 2000, to 14.5 percent in 2014.
Switzerland has twice hosted the Olympics, in St Moritz in 1928 and 1948.
It has a long association with the Olympic movement, since the IOC's headquarters have been based in Lausanne for the past century.