The Navajo Nation filed a claim Monday seeking more than $160 million from the federal government for damages tied to last year’s Gold King Mine disaster in Colorado, which sent three million gallons of toxic sludge into nearby waterways and triggered environmental concerns in several states.

In a letter to the Environmental Protection Agency, the Navajo Nation requested $3.2 million to cover costs already incurred as a result of the spill, and another $159 million to pay for ongoing environmental monitoring and an alternative...