The governors of at least four states expressed initial interest in an offer on Thursday from Interior Secretary Sally Jewell to allow them to reopen national parks in their states, though it remained unclear whether any such deals could be reached soon.

After more than a week of requests from several states that they be allowed to reopen parks closed by the federal government shutdown, Ms. Jewell said the administration would be willing to consider such agreements as long as the states agreed to pay the full operating costs, including the salaries of all federal park employees.

“South Dakota, Utah, Colorado and Arizona have all expressed some initial interest in exploring a potential agreement,” said Blake Androff, chief spokesman for the Interior Department in Washington. “But discussions are in the early stages. No formal offers have been submitted.”

In Utah, officials hailed the offer as a breakthrough that could defuse rising tensions over the closing of the state’s five national parks. Utah’s governor, Gary R. Herbert, has urged the Interior Department to let the state use its own money to reopen them.