WASHINGTON—Special counsel Robert Mueller told Attorney General William Barr in March that his four-page letter summarizing the findings of the Russia investigation failed to “fully capture” the team’s work and threatened the public’s confidence in the results of the probe, according to a person familiar with the matter.

Mr. Mueller stated his concerns in a letter he sent March 27, after Mr. Barr publicly announced that the special counsel’s investigation hadn’t found that the Trump campaign conspired with Russia to interfere...