Whatever their purpose, Mr. Conway’s messages have caught the attention of aides in the West Wing, according to a White House official. That official said some advisers were miffed by the tweeting, and considered Mr. Conway’s public criticisms to be potentially harmful to Ms. Conway, who remains poised to assume more of the duties left behind by Hope Hicks, the departed White House communications director.

But the president does not seem to share those concerns, according to a senior White House official familiar with his thinking. He continues to trust Ms. Conway, and he appeared to enjoy her response after the CNN journalist Dana Bash recently asked her to explain her husband’s social media missives.

“It was meant to harass and embarrass,” Ms. Conway said of the question. She said that there had “been a different standard” for her than others when it came to scrutinizing her husband’s opinions, and she chided Ms. Bash for pressing about “how people’s spouses and significant others may differ with them.”