WASHINGTON—President Donald Trump is considering a major shake-up of his staff, a senior administration official said Friday, as he looks to build a more cohesive operation after a tumultuous start in the White House.

As he turned to the Syrian crisis that prompted the White House to adopt a war footing culminating in Thursday’s missile strike, Mr. Trump was increasingly unhappy over the feuding that has run through the West Wing and is determined to see it end, the official said.

Mr. Trump is specifically evaluating whether to keep his chief of staff, Reince Priebus, and his chief strategist, Steve Bannon, in their current positions.

Two people close to the White House said Mr. Trump has been talking to confidants about Mr. Priebus’s performance and has asked for names of possible replacements.

Mr. Trump is “trying out different names with his friends,” one of those people said.