On a leaked conference call, leaders of dark-money groups and an aide to Mitch McConnell expressed frustration with the popularity of the legislation—even among Republican voters.

In public, Republicans have denounced Democrats’ ambitious electoral-reform bill, the For the People Act, as an unpopular partisan ploy. In a contentious Senate committee hearing last week, Senator Ted Cruz, of Texas, slammed the proposal, which aims to expand voting rights and curb the influence of money in politics, as “a brazen and shameless power grab by Democrats.” But behind closed doors Republicans speak differently about the legislation, which is also known as House Resolution 1 and Senate Bill 1. They admit the lesser-known provisions in the bill that limit secret campaign spending are overwhelmingly popular across the political spectrum. In private, they concede their own polling shows that no message they can devise effectively counters the argument that billionaires should be prevented from buying elections.