New York City is struggling to get an accurate count of the number of Covid-19 fatalities as more people die at home and some likely coronavirus deaths are attributed to heart attacks and other causes, Mayor Bill de Blasio acknowledged Wednesday.

On Tuesday, New York City officials said they would begin counting "probable" Covid-19 deaths, which are people "who had no known positive laboratory test," but are believed to have died due to Covid-19. The New York City Department of Health and Mental Hygiene estimates that there have been 3,778 probable Covid-19 deaths since March 11 that weren't previously counted in the city's official tally. There have been 6,589 confirmed Covid-19 deaths in New York City so far.

"I want the whole truth out. Wherever the facts take us, I want the whole truth out," de Blasio said Wednesday at a news briefing. "Absolutely, I believe there are more people who died because of Covid-19, in one way or another, because of something that happened to them related to Covid-19."

New York City has had to prioritize coronavirus testing due to a scarcity of test kits and the supplies needed to analyze them, de Blasio has previously said. As a result, testing has gone largely to those who are already in the hospital and first responders, according to de Blasio. People exhibiting symptoms consistent with Covid-19 but who remain at home may not be tested and thus excluded from the official count of cases — even if they die.

De blasio has repeatedly emphasized that the city is operating on a "wartime dynamic," meaning that officials are making decisions rapidly as the situation evolves. He said the decision to report probable Covid-19 deaths was made to be more transparent with the public about the magnitude of the outbreak.

NYC Health Department Commissioner Dr. Oxiris Barbot said most people who have died from Covid-19 were diagnosed in hospitals. However, the city has seen significantly more people dying at home than usual, Barbot said.

Barbot said the city has compared "the number of deaths during that same time period in the previous year and what we find is that there are roughly 3,000 deaths above what would have been anticipated."

In the first five days of April, the number of people who died in their homes or on the street in New York City was more than eight times the deaths recorded during the same period last year, The New York Times reported, citing the New York Fire Department.

The official count could also be missing patients who suffered from other illnesses. Barbot called for more research into how Covid-19 interacts with other illnesses. For example, she said some people who have died after having a heart attack but also exhibited Covid-19 symptoms might be missed in the official tally.

"Are there potentially cases where someone registered a death as a heart attack because the person hadn't yet developed symptoms of Covid-19 and truly should have been categorized as a Covid probable?" she said. "I don't know if we'll ever be able to answer that question fully, but I think it's something that still needs to be looked into further."

Dr. J Mocco, Director of Mt. Sinai's Cerebrovascular Center in New York City, which handles stroke patients, told CNBC that there's an emerging base of evidence to indicate that Covid-19 exacerbates strokes and potentially other medical conditions.

"We're seeing more strokes than before and they are a large percent Covid positive," he said. "We had five patients over a two-week time period who were 49 years old or younger, which is not standard, who had no other risk factors except for being Covid positive."

Mocco said it's possible that people are suffering strokes and staying at home out of fear of catching Covid-19 in the hospital, but they might already have the disease and it could even be a cause of the stroke. Such patients would not be tested for Covid-19 and thus could be missed by the city's official tally. Mocco's team at Mt. Sinai, in the epicenter of the U.S. Covid-19 outbreak, continues to research how the respiratory disease interacts with other illnesses.

— CNBC's Noah Higgins-Dunn and Berkeley Lovelace contributed to this report.