Jennifer Nuzzo, a senior scholar at the Johns Hopkins Center for Health Security, which is closely tracking the coronavirus pandemic, said that “the officially reported numbers don’t reflect the true level of illness and death that have occurred.”

“We very much feel the reported numbers reflect an undercount,” she said.

Inside the White House, doubts about the official numbers are pervasive, though they come in different forms. Mr. Trump is in search of good news to promote his administration’s response to the pandemic and to press states to reopen. Dr. Deborah L. Birx, the White House’s coronavirus response coordinator, is a numbers obsessive and wants her own data to supplement information coming in from the states and the C.D.C. One official has even accused hospitals of potentially exaggerating their coronavirus patient counts to milk money from Medicare.

Top White House officials have even discussed appointing a “forensic” team to audit how some hospital systems and state health departments have been tallying infections and deaths, according to one senior administration official. Multiple senior White House officials said they were unaware of such conversations.