WASHINGTON—Former Deputy FBI Director Andrew McCabe approved investigations into President Trump’s potential ties to Russia almost immediately after the president fired his former boss in May 2017, Mr. McCabe said in his first public interview since that tumultuous month.

In a clip from CBS’s “60 Minutes” posted Thursday, Mr. McCabe said he authorized both counterintelligence and criminal inquiries into Mr. Trump to give the cases solid footing—in case he was fired or reassigned—two days after Mr. Trump fired Federal Bureau...