WASHINGTON—President Donald Trump’s newly appointed communications director acknowledged his power struggle with the White House chief of staff and said their differences may be unbridgeable, using profane language in published comments to express his frustration with his new colleagues.

Anthony Scaramucci, who joined the White House last week and reports directly to the president rather than to the chief of staff, as is customary, said he and Chief of Staff Reince Priebus have been at odds.

“We...