Organising chief Giovanni Malago told an International Olympic Committee Session in Mumbai in October his organisation had been effectively ordered by the Italian government to move the bobsleigh, luge and skeleton competitions due to spiralling costs.
Malago had said the events would likely be held outside Italy.
But Salvini has since said the ministry of transport would make a fresh proposal for a new track to be built at Cortina d'Ampezzo, which was the original plan.
"The Milan-Cortina Olympic Games must be an Italian Olympic Games," Salvini said on the sidelines of a trip to Milan.
"This is what we're working on at the ministry, without spending another euro or wasting any more time...
"These 2026 Winter Games are an incredible opportunity in terms of development and image not only for Lombardy and Veneto, but for all of Italy.
"The engineers told me that we still have enough time to build this track, we're working on it."
The organising committee has said the deadline for making a decision on the track's location has been agreed with the IOC for the end of January 2024.
The organisers also said earlier this month that they had contacted "the American, German, Austrian and Swiss Olympic committees" in case the unprecedented decision of relocating the events abroad was reached.
Renovating the now disused track, which was built for the 2006 Turin Games in Cesana, has been touted as a possible third option.
