In a striking contradiction from the White House statements about coronavirus being under control, the Pentagon’s chief of personnel is sharply warning that the infection poses an increased “threat” in areas where US troops and defense personnel are located around the globe.

In a Tuesday memo obtained by CNN, Matthew Donovan, who is filling the top personnel job on a temporary basis, told the Chairman of the Joint Chiefs, all of the services and other top Pentagon offices that as the novel coronavirus disease (COVID-19) continues to spread it is “an increasing force health protection (FHP) threat in areas where Department of Defense (DoD) personnel live and work.”

“Force health protection” refers to the various levels of risk of infection and the as that risk rises, the measures commanders may take to protect their troops.

CNN has learned that top US commanders around the globe are increasing concerned that as the impact of the virus spread and allies shut down borders and travel, that US military readiness may begin to be at risk as soon as a month from now, according to several defense officials CNN has spoken to.

Where things stand now: A key exercise has already been cancelled in South Korea. More than 60 US personnel who traveled to Israel for an exercise to train against regional threats such as Iran, began returning to their bases in Europe Friday at the request of the Israeli government although none of them were reported to be ill. And the US European Command is now monitoring the infection’s impact to see if it's able to continue a lengthy upcoming exercise with upwards of 20,000 troops training against a Russian threat.

Defense officials say as long planned US military exercises are postponed or cancelled, troops will be less up to date on their critical training for future operations. But in addition, the spring and summer is a regular time frame for thousands of personnel to routinely change assignments worldwide. If they and their families cannot readily travel overseas to new jobs, and currently overseas troops cannot rotate, this can lead to further turmoil and uncertainty in the military personnel system officials say.

Publicly, Pentagon officials are not discussing details of how training and operations could be impacted. On Wednesday Defense Secretary Mark Esper told the House Armed Services committee that he has not yet had discussions about whether or not the Defense Department would need additional funds from Congress to meet the coronavirus threat, and would need to have that discussion with the combatant commanders first.

Chairman of the Joint Chiefs of Staff Gen. Mark Milley assured the committee that the department is “taking all the appropriate measures right now,” while adding, “we’re doing the estimates of the situation, so we owe you some answers.”

But field commanders are not waiting to request additional supplies. Additional doctors, nurses and medical technicians are expected to arrive at US military installations in South Korea in the coming days. At a US army base in Vicenza in northern Italy, extra package field rations are being shipped in for possible use if large numbers of troops are quarantined on base.