The Federal Council said Sion 2026 offered "a great opportunity for sport, the economy and society" in the wealthy alpine nation.
It offered a total of 995 million Swiss francs ($1 billion, 862 million euros) to support the Games in the Valais city.
Sion has repeatedly been listed as a candidate to host 2026, with interested cities due to formally unveil their projects in 2019.
But the International Olympic Committee has faced challenges in attracting viable candidates, as voters increasingly grow weary of the cost.
Swiss federal government support for a bid will be welcome news at IOC headquarters in Lausanne, especially after the Austrian state of Tyrol voted against Innsbruck's bid for 2026, citing concerns over the financial burden and environmental impact.
After reviewing a feasibility study for Sion 2026, the Federal Council praised plans "to rely as much as possible on existing infrastructure" and its commitment to "sustainable development".
The government offered eight million francs to support the bid process and another 827 million to the hosting, among other contributions.
The Federal Council underscored that a successful Swiss hosting effort will require support at the cantonal level, a complicated issue in a country where major issues are often decided by public ballot.
Voters in the canton of Graubünden rejected a Winter Games bid proposal in a February referendum.
Sion 2026 could also collapse if voters in Valais reject the idea.
On Wednesday the Valais government expressed its delight at the Federal Council's support and said the population would be called on to vote on the issue on June 10th next year.
Some 60 million francs would have to be invested in existing infrastructures in order to host the Games, said the canton in a statement.
Sion is also counting on the neighbouring cantons of Bern, Fribourg and Vaud to host events, meaning four separate referenda may be needed before the plan can move forward.
The committee for Sion 2026 on Wednesday rejected hundreds of thousands of francs offered by FC Sion boss Christian Constantin in support of a bid, wrote news agency ATS.
Committee president Jean-Philippe Rochat said Constantin, who has been a vocal supporter of Sion 2026, was no longer involved in the bid because of an incident in Lugano in which he hit former Swiss national team coach and TV pundit Rolf Fringer.
Last week Constantin was banned from football for 14 months over the incident.
Next year's Winter Olympics will be held in Pyeongchang, South Korea, while Beijing hosts in 2022.