WASHINGTON — With tensions escalating between President-elect Donald J. Trump and prominent black leaders, Mr. Trump met privately on Monday with the eldest son of the Rev. Dr. Martin Luther King Jr. on the holiday devoted to the civil rights hero.

The hastily arranged meeting at Trump Tower occurred as Mr. Trump feuded publicly with Representative John Lewis, Democrat of Georgia, who fought for civil rights alongside Dr. King.

It has highlighted the challenges Mr. Trump faces as he prepares to take office on Friday deeply distrusted by minorities across the country, many of whom have been offended by his false allegations that President Obama was born outside the United States, appalled that his candidacy drew backing from white supremacist organizations, and dismayed at policy proposals they consider antithetical to their interests.

Mr. Trump on Monday did not address those issues, appearing in the lobby to allow news cameras to capture pictures of him shaking hands with Martin Luther King III, but ignoring questions shouted by reporters about their conversation or his statements about Mr. Lewis.