Louisiana has only 381 ICU beds available, Edwards said, and he expects the need will overwhelm the supply in hospitals by early April.
"We believe by April 7 we will lack capacity in our hospitals," Edwards told the president.
Trump approved the major disaster declaration for the state on Tuesday.
"You got hit, a delayed hit," Trump told Edwards, according to detailed notes from the call obtained by ABC News. "Looked like you weren’t going to be hit at all, and it sort of exploded."
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New Orleans and, by extension, Louisiana -- a city and a state that are no strangers to sudden, life-changing disasters -- are en route to joining New York as another epicenter of the coronavirus outbreak. As the state awaits federal assistance, Edwards is doing what he can to prepare.
During a press conference on Thursday, Edwards said that the hospitals around the state would be increasing their ICU bed capacity by several hundred.
Morial Convention Center in New Orleans will be used as a medical facility to accommodate the expected hospital overflow, with 1,120 beds expected to arrive over the weekend.
And Edwards has been seeking support from a Veterans Affairs hospital in the state. Trump asked about the progress of that effort during the call with governors on Thursday.
"If you have a problem, let me know," Trump said. "You can call me directly."
Doctors and nurses from across the state have detailed to ABC News their experiences from inside hospitals that are facing significant shortages of personal protective equipment, much like the rest of the country.
Dr. John Dwyer, co-chair of Infection Control at Tulane Medical Center, said the hospitals have started to ration, disinfect and reuse personal protective equipment (PPE) in order to conserve supplies -- a decision he said "felt horrible."
"We have a lot of nurses and physicians who are really working hard and putting themselves at risk and it’s a huge source of anxiety for many people," Dwyer told ABC News. "I have lost sleep about it. A lot of people have."
"We can put a man on the moon, but the government can’t put a face mask on our doctors," Dwyer continued. "How can we not have the resources and the manpower to provide for our health care workers?"
To try and combat the supply shortage until a larger government response is available, Dwyer has spearheaded a PPE donation effort alongside a pair of medical students.
"We wanted to help out before the [the hospitals] got to a dire point," Alex Woodbridge, a third-year medical student and president of the Student Clinic Council, told ABC News.
Since launching the effort last week, the group -- with the help of about 100 additional volunteers -- has already managed to collect thousands of gloves, gowns, N95 face masks, and other supplies, which they have started to deliver to hospitals.
"A lot of manufacturers in the country have ramped up and we hope that can meet the need," Sophie Foroushani, another third-year medical student, told ABC News. "But we are trying to bridge the cap until supplies are available in large quantities."
In Baton Rouge, the chief operating officer of Baton Rouge General Hospital, Stephen Mumford, said they have been preparing for the outbreak for several weeks. The hospital has been working with the state and receiving PPE donations from community organizations, companies and individuals, but officials are still concerned about making the supply last.
"We just don’t know how long this is going to last," Mumford said. "We’re really trying to preserve what we have and make sure that we’re appropriately using it so our caregivers, two to three months from now when we’re still in the middle of it, will have masks that are going to be critical for their protection."
Mumford said the hospital has been under a big financial strain as it cuts down on lucrative elective surgeries and focuses on COVID-19 response and other essential care, but among the hospital’s biggest challenge has been the anxiety of their staff and caregivers.
"We’re really working hard to over communicate with our staff and reassure them that if they properly protect themselves, they’ll be okay and that it’s okay to go home to your families," Mumford said. "And our staff has done a phenomenal job keeping ourselves safe and being able to take care of our patients."
But fear and anxiety seem inevitable for nurses and hospital staff jumping into the front lines with limited resources.
Louisiana State Nurses Association Executive Director Thea Ducrow, who conducted a survey with 701 nurses across the state, said nurses and other health care workers in the state are being pushed to dangerous situations without proper preparation and protection.
The survey showed that the single biggest concern for nurses at the moment, Ducrow said, is the shortage of personal protective equipment. Several nurses are being asked to reuse items, she said, and are being thrown into patients’ rooms without properly fitted masks and little guidance from their employers on how to protect themselves for potential exposure as they care for COVID-19 patients.
"For years, they were told this is how to do the infection control and they were given explicit examples and procedures to follow," Ducrow said. "And now they’re being asked to do something different."
One Louisiana nurse who spoke with ABC News on the condition of anonymity said that her hospital has been rationing and reusing one box of "flimsy masks" and using bleach wipes to clean the hospital because sanitizing wipes are on backorder.
"I think I speak for everyone when I say that this is very scary and unsettling," the nurse told ABC News. "Never have I experienced such panic, shortage of necessary PPE and sanitation supplies at work."
"The stress is unimaginable," the nurse added. "We're all scared of what's to come."