Jared Evans, a member of the Indianapolis City-County Council, is proud that the city is among 20 finalists for one of the most coveted prizes in the country: the planned second headquarters of Amazon.

He does, however, have one small question: What financial incentives did his city dangle in front of Amazon?

“What have I been told?” Mr. Evans said. “Absolutely nothing.”

Across the country, the search for HQ2, as the project has been nicknamed, is shrouded in secrecy. Even civic leaders can’t find out what sort of tax credits and other inducements have been promised to Amazon. And there is a growing legal push to find out, because taxpayers could get saddled with a huge bill and have little chance to stop it.

“The only time the public may become aware if the city has promised Amazon incentives is if we win and then we need to get those incentives passed,” Mr. Evans said.