Facebook Inc.’s senior leadership is increasingly divided over how to address criticism of the company’s effect on U.S. politics, with board member and billionaire investor Peter Thiel serving as an influential voice advising CEO Mark Zuckerberg not to bow to public pressure, according to people familiar with the matter.

One flashpoint of late: political advertisements. Mr. Thiel has argued that Facebook should stick to its controversial decision, announced in September, to continue accepting them and to not fact-check those...