Some of Mr. Trump’s closest advisers are adamant that the best way forward is to downplay the dangers of the disease. Mark Meadows, the chief of staff, has been particularly forceful in his view that the White House should avoid drawing attention to the virus, according to people familiar with the discussions.

Mr. Meadows has for the most part opposed any briefings about the virus, while other Trump advisers, including Hope Hicks and Jared Kushner, have been open to holding briefings so long as they are not at the White House — where Mr. Trump could show up and commandeer them. Mr. Pence’s team would like to hold more briefings with the health experts, but some of Mr. Trump’s communications aides do not want the vice president to be part of them.