ALEXANDRIA, Va.—Paul Manafort is now set to face a jury next week in the first trial from special counsel Robert Mueller’s investigation, after a federal judge agreed to push the trial back several days so the defense team could finish going through evidence.

“There are equities and good reasons on both sides of this motion,” U.S. District Judge T.S. Ellis said in moving the trial of President Donald Trump’s former campaign chairman to July 31 from its previously scheduled Wednesday start.

Mr....