WASHINGTON—In January, President Donald Trump’s legal team and special counsel Robert Mueller were zeroing in on the terms of a presidential interview: the last Saturday of the month, at Camp David, on a narrow list of topics, for maybe between two and six hours, with time set aside for bathroom breaks, according to people familiar with the matter.

The lawyers presented the terms to the president, who was described by one person as “optimistic” about the prospect. Ultimately, though, the deal was never sealed as members of...