Hospitals have begun immunizations for health care workers, following the recommendations made for swine flu, but Cooper said that since the first vaccines are available in the nasal mist form, giving it to health care workers is the only proper way to go.
"What we know is this nasal mist that's been distributed cannot be given to pregnant women, or children or adults with chronic disease," she said. Those are groups that public health officials say should also be high on the priority list for vaccination.
"Health care workers certainly need to be at the front of the line, because if any Tennessean gets sick with H1N1 or has a heart attack or has any condition" that requires medical attention, "we want to make sure that the health care work force is sufficient to take care of their needs."
"My sense is people understand that health care professionals will be the ones taking care of people, so it's important for them to be immunized," said Michelle O'Keefe, director of public affairs for Wishard. "We want our employees to be here and be well, and I get the sense that people understand that."
Hospital officials said they could not predict what might happen as vaccines become available to members of the general public.
Sara Burnett, a spokeswoman for Le Bonheur, said that while hospital workers have begun to receive the vaccine, they have also received inquiries from people who cannot yet get vaccinated but want to.
"We've had questions about when is it available," she said. "We've referred those back to the Tennessee Department of Health."
And some primary care doctors, responding to an ABC News inquiry, said they have not yet received the vaccine, but they have gotten plenty of inquiries.
"I'm getting pounded," said Dr. Randy Wexler, assistant professor of clinical family medicine at Ohio State University Medical Center. "People want to know when they can get it, and they want to be placed at the top' of the waiting list."
A similar rush is taking place in Washington, D.C.
"Yes, patients are asking when the H1N1 flu vaccine will be available," said Geeta Nayyar, an assistant clinical professor of medicine at George Washington University. "So much so, that they are trying to coordinate their office visits with when the vaccine is available, so they can make one office visit."
But while many people may be asking for vaccinations now, Tennessee's Cooper said she has chosen to look on the bright side.
"This is just the beginning," she said. "So again, we will continue to see vaccine being delivered to our state and hopefully within a very few weeks there will be enough vaccine for any Tennessean who wants it to get it."
Doctors nationwide are hoping for the same thing.