The request comes amid a push by Interior Secretary Ken Salazar to find more places suitable for bison on public and tribal lands across the West.
Montana Gov. Brian Schweitzer has also sought new habitat for the animals, although that effort has run up against determined opposition from the livestock industry.
They asked the state to transfer an unspecified number of the approximately 170 bison held on Turner's Green Ranch near Bozeman to the Wind River Reservation in western Wyoming.
The tribes also requested help in their effort from Salazar. His office did not immediately respond to a request for comment.
Yellowstone has a burgeoning population of more than 4,200 bison that periodically spill into surrounding areas of Montana. Thousands have been captured and sent to slaughter in the past two decades to avoid spreading the disease brucellosis to livestock.
The bison on Turner's ranch were spared when they were put into a government quarantine program for several years to ensure they were disease-free. They had been slated to go to the Wind River Reservation several years ago but ended up on Turner's ranch when the tribes' proposal fell through.
"Our ultimate goal is to have a free-roaming herd to some extent that exists on the reservation," said Jason Baldes, a member of the Eastern Shoshone who said he's been working with government agencies and conservation groups to move the animals onto the reservation.
"There are a lot of things to work out, such as management, once we get the buffalo on the ground," Baldes added.
Bison once numbered in the tens of the millions across the West before overhunting in the late 1800s nearly drove them to extinction. Yellowstone has one of the largest remaining wild populations.
Another group of quarantined animals from the park was moved this year to northeast Montana's Fort Peck Indian Reservation. A lawsuit from ranchers, property rights advocates and others blocked plans to move some of the animals onto a second reservation.
Schweitzer, a Democrat who has championed the cause of bison restoration to the consternation of the cattle industry, has sought to get the Green Ranch bison onto the National Bison Range in Moise.
Salazar in May asked the heads of the U.S. Fish and Wildlife Service, National Park Service and other agencies to "examine the feasibility of the relocation of (Yellowstone) bison to suitable federal or tribal lands."
A representative of a conservation group that has been working with the Wind River tribes said the reservation is the most suitable destination for the animals since some of the groundwork for a relocation already has been accomplished.
But Garrit Voggesser with the National Wildlife Federation said the Green Ranch bison also could be broken up and sent to more than one destination.
"If we just keep taking these steps, ultimately we'll get to the point where we have bison on a much broader landscape," said Voggesser.
-- The Associated Press