With six months to go until November’s midterm elections, Mr. Mueller’s investigation into Russian meddling in the 2016 presidential campaign will soon run into a dead zone of sorts, in which former prosecutors say they expect him either to wrap up, or lie low and take no visible steps until after the election.

Though Mr. Mueller doesn’t face any specific legal deadline, the fall midterms amount to a political one, according to experts and prosecutors. He will reach a point this summer when Justice Department habits dictate that he will have to either finish his inquiries or go dark and stretch out his work until past November so he doesn’t appear to be trying to sway voters’ decisions, which would be at odds with Justice Department guidelines for prosecutors.

Mr. Mueller has a lot still to do—prepare several reports, bring expected charges against alleged Russian hackers behind the breach of the Democratic National Committee and make decisions on whether to prosecute other cases. Perhaps the most politically sensitive issue he has yet to resolve is whether the special counsel’s office will interview President Donald Trump.