Originally Published: 26 APR 21 15:44 ET

By Scottie Andrew, CNN

(CNN) -- A hospital system in Houston is requiring all of its employees to get vaccinated against Covid-19, making it one of the first major hospital systems in the US to mandate vaccination among employees and move to fire them if they don't comply.

Houston Methodist, a network of eight hospitals that has 26,000 employees, said it will require every employee to provide proof of vaccination by June 7.

If employees aren't vaccinated before the June deadline, they'll be suspended, without pay, for two weeks. If they're not vaccinated within that suspension period, the company will "initiate the employee termination process," according to the company's new HR policy, implemented this month.

"As health care workers we must do everything possible to keep our patients safe and at the center of everything we do," Houston Methodist CEO Dr. Marc Bloom told employees in an email obtained by CNN. "By choosing to be vaccinated, you are leaders -- showing our colleagues in health care what must be done to protect our patients, ourselves, our families and our communities."

Employees have until May 3 to apply for exemption for religious or medical reasons. If their request is denied, they're expected to get vaccinated before the June deadline.

The hospital system already mandated the vaccine for "Phase 1 employees," which included new hires, executives and managers. They were expected to provide proof of vaccination by mid-April.

About 89% of all employees have been vaccinated so far, Bloom said in the email.

The hospital system previously offered employees $500 to get their Covid-19 vaccine, CNN reported in January.

Other hospital systems in area may follow suit

Houston Methodist's decision upset some employees who said they're not comfortable getting vaccinated.

Of those 1,200 employees in the first phase of required vaccinations, two "chose to leave the organization," Bloom wrote in the email to employees.

"We are sorry that they made that choice, but by doing so, they are putting themselves before the safety of our patients, which is not consistent with our culture," he said. "By getting to 100%, we can show our community and our patients how much we care about them."

At least two other major hospital system in the area may follow suit.

In an email, Dr. James McDeavitt, Baylor College of Medicine senior vice president and dean of clinical affairs, told CNN that the college is considering mandatory vaccination among its employees, too.

Memorial Hermann Health System, a non-profit hospital system that serves greater Houston, will mandate vaccines for its employees, too, the health system said in a statement to CNN.

There's no deadline for vaccination yet, but it will occur around the time its hospitals relax some other Covid-19 prevention measures like social distancing and masking, the statement read.

Houston Methodist, Baylor College of Medicine and Memorial Hermann all require employees to get vaccinated against other infectious diseases, including seasonal flu, hospital leaders told CNN and Houston's Chron.com has reported.

Several colleges, including Duke, New York University and schools within the University of California system, are mandating vaccines for students to return to campus in the fall. Public and private universities already require students to be vaccinated against diseases like mumps and measles.