Ms. Brzezinski was expected to address the incident on-air Friday morning. For now, her only public comment has been a photograph she posted to Twitter of a box of Cheerios, emblazoned with the words, “Made for Little Hands,” referring to a longstanding insult about Mr. Trump.

CNN, one of MSNBC’s chief rivals, issued a statement in support of Ms. Brzezinski — a rare sign of solidarity in the famously fractious TV news world. The message spoke to a growing sense among journalists that they must unite against the anti-press barrage from Mr. Trump and his aides, particularly after a week where Ms. Sanders, at the White House lectern, encouraged Americans to watch an anti-CNN video, “whether it’s accurate or not.”

At her televised briefing Thursday, Ms. Sanders faced tough questions about the tweets from numerous outlets, including two journalists for Fox News, John Roberts and Jon Decker, who posed some of the session’s more pointed queries.

“The only person that I see a war on is this president and everybody that works for him,” Ms. Sanders said, echoing her comments earlier this week that Americans “deserve something better” from the news media.

“I don’t think you can expect someone to be personally attacked day after day, minute by minute, and sit back,” Ms. Sanders told the journalists. “The American people elected a fighter; they didn’t elect somebody to sit back and do nothing. They knew what they were getting when they voted for Donald Trump.”

That desire to punch back has fueled several of the president’s media feuds, including his ongoing clash with CNN, whose president, Jeffrey A. Zucker, hired him to host “The Apprentice” at NBC.

The president has bragged to friends that he secured Mr. Zucker’s current job by recommending him to a top executive at Turner, CNN’s parent company. People familiar with CNN’s hiring process have said Mr. Trump’s endorsement did not play a role.