I wish that such a trail existed. 100 years ago one could bicycle anywhere in the park. Now it is not only forbidden, but possibly very dangerous. For now, other than a couple short trails, we have to be content with riding the roads in and through the park. Fortunately the speed limit is 45, making the roads relatively safe.
East of the park there is some riding in Wyoming. Grizzly bears are an issue. North of the park is the Gallatin Crest, temporarily closed due to an environmental lawsuit. It was about a 1-1/2 day ride. Northwest of the park is a good one day ride from Hebgen Lake to Taylor Fork. West of the park is a good quality segment of Continental Divide Trail south of the town of West Yellowstone. Just north of this is the endangered area called Lionhead. Very good riding here, some grizzlies though. Further west, about 20 miles from the park, is the Centennial Range, with about 3 days of riding along the crest. North of the Centennials is the Gravelly Range. This has a gravel road along the crest, with ATV trails leading off of it, and apparently a couple singletrack trails that I am not familiar with.
I guess that you either heard about the Gallatin Range or the Centennial Range.