WASHINGTON—Donald Trump’s former campaign chairman and two other Trump associates have agreed to an interview with the House committee probing alleged Russian interference in the 2016 election, as a partisan divide and the actions of the panel’s leader threatened to derail the inquiry.

Paul Manafort, who served as Mr. Trump’s campaign chairman and ran the day-to-day operations of the campaign for about five months, agreed through an attorney to speak with the House Intelligence Committee, the panel’s chairman announced Friday.

...