WASHINGTON — Aged nuclear plants in Vermont and Illinois may be playing the equivalent of musical chairs in a graveyard, vying for space at a dump in Texas whose owner hopes to accept radioactive waste from many other states.

Under an alliance struck 16 years ago between Vermont and Texas, tiny Vermont can fill up to 20 percent of the space at any low-level nuclear waste dump built in Texas’ wide-open spaces. Texas got the right to exclude other states’ waste. But as a company prepares to begin construction this month on the state’s first one, the arrangement may be jeopardized by swiftly changing circumstances.

A private company that won a contract to operate the plant, at a site in Andrews on the New Mexico border, wants to accept waste from the 36 states that do not have access to a dump for some of their waste now. And a commission made up of representatives from the two states that controls the planned dump has proposed a rule for accomplishing that.

Waste disposal is so difficult, says the company, Waste Control Services, that power plants and other generating sources have reduced their volumes sharply. And Vermont and Texas together produce so little that, the company adds, it would have to charge huge amounts per cubic foot and per unit of radioactivity to get its investment back.