President Donald Trump said he would be making a “big announcement” next week about his proposal for overhauling the tax code, but his budget director said that it might be June before the White House releases its detailed plan, and that even broad principles could be weeks away.

In brief remarks while signing executive orders on Friday, Mr. Trump didn’t go into any specifics about a tax proposal that was one of the central promises of his 2016 presidential bid. But with the 100-day milestone arriving on April 29, the White...