The man tapped to run the Labor Department isn’t sure the agency should be using workers’ pay levels to determine if they’re automatically eligible for overtime compensation.

An Obama administration rule to make some 4 million workers newly eligible for overtime pay—by doubling the salary threshold for automatic eligibility to $47,500—is on hold, pending federal litigation in Texas. Although the rule has sparked a debate over where to set the salary threshold, labor secretary nominee Alexander Acosta noted that the judge in that case has raised questions about whether the DOL should instead focus on the actual duties workers perform....