WASHINGTON—White House Counsel Don McGahn this summer was so frustrated about the lack of protocols surrounding meetings between President Donald Trump and Jared Kushner, his son-in-law whose activities are under scrutiny in the Russia probe, that West Wing officials expressed concerns the top lawyer would quit, according to people familiar with the conversations.

Mr. McGahn expressed concern that meetings between Mr. Kushner and Mr. Trump could be construed by investigators as an effort to coordinate their stories, three people familiar the matter said.

Two senior White House officials—then-Chief of Staff Reince Priebus and former chief strategist Steve Bannon—urged Mr. McGahn not to resign, according to people familiar with the conversations. One person characterized Mr. McGahn’s frustration as, “Fine, you’re not taking my advice? Why stay?”

Mr. McGahn stayed in the job, reassured in part by the White House’s hiring of a legal team specifically to manage the response to the probe of Russian meddling in the 2016 election. Attorney Ty Cobb was hired to lead that group.