Donald Trump and his then-attorney Michael Cohen pressured CNBC in 2014 to place the real-estate tycoon higher in its list of the country’s top business leaders after Mr. Cohen failed to manipulate the rankings in Mr. Trump’s favor, according to people familiar with the matter.

Mr. Cohen called CNBC and threatened that Mr. Trump would sue over his poor standing in the ranking, arguing that the news channel was “ignoring the will of the people,” the people familiar with the matter said. CNBC didn’t respond to the threat, and...