"They're letting us down with no protection," Tennison said. "They always say that we're on the front lines of security, and if we're on the front lines like the military, we need to have our equipment."
Still, he said, employees have been advised to stay home if they feel ill and to take precautions.
At Los Angeles International Airport, TSA employee Bobby Orozco Jr. said he's heard from his co-workers that supervisors have said employees are not allowed to wear masks, but he hasn't seen any firsthand information on that.
Even so, Orozco said there are no masks available to put on even if he and his co-workers wanted to do so. He said such decisions often are up to federal security directors at individual airports.
Orozco said he and his peers are "visibly and obviously upset because you see passengers coming through with them on. And it's almost like TSA is saying the passengers are going to be afraid, we don't want to cause the panic -- but the panic's already here."
Orozco also said employees can't bring their own masks because, "If they're not TSA-issued, they're not going to approve them anyway."
Today, T.J. Bonner, president of the union National Border Patrol Council, told ABC News he has not seen the same situation among border patrol agents. Bonner said he has inquired and not found a single incident of border patrol agents being asked not to don protective gear.
The National Border Patrol Council issued an alert to employees Monday, advising them to call in sick if needed, to take precautions recommended by the CDC when interacting with potentially sick people and to speak up "if management refuses to issue N95 respirators or forbids you from using them."
Today, an official with Customs and Border Protection said employees are provided with protective equipment including gloves and masks, which are readily available to use on the job.
On Sunday, Napolitano likewise said, "The CBP is inventorying for every duty station and every employee our resources -- personal protective equipment, and so forth -- to make sure that we have adequate supplies on hand at the borders themselves."
The American Federation of Government Employees sent a letter to TSA leadership Wednesday on the subject.
"We have learned that TSA management at U.S. airports are denying the use of respirators and gloves to TSOs under most circumstances," the letter said. "TSOs have been told that they cannot wear respirators because doing so would alarm the public."
The letter requested that protective gear be made available to any worker who requested it, and also sought other actions intended to better prevent the spread of the virus.
The suggestion that U.S. health officials are preventing those monitoring ports of entry from protecting their personal health comes on the same morning that Vice President Joe Biden made waves for suggesting Americans should avoid mass transit to guard against swine flu.
The vice president's comments have caused a stir for elevating concerns to a level far beyond that disseminated by the government.
Meantime, the World Health Organization has confirmed that seven people in Mexico have died from the H1N1 flu virus, in addition to one person who died yesterday in the United States after arriving from Mexico.