ATLANTA — President Trump has been open about his eagerness to see the economy and some semblance of business as usual spring back to life as soon as possible. His surgeon general, Jerome Adams, in a television interview on Friday noted the potential for reopening the country — “place by place, bit by bit,” beginning as early as next month.

But on Sunday, officials still in the thick of the grim reality caused by the coronavirus pandemic urged caution, fearing that relaxing protective measures too early could cause the virus to surge once again.

In interviews on Sunday morning talk shows, governors and mayors acknowledged the delicate balance between aggressively combating the virus and limiting the economic pain, but they said that public health concerns were their priority.

“We could be pouring gas on the fire, even inadvertently,” Gov. Philip D. Murphy of New Jersey, a Democrat, said in an interview with CNN on Sunday. He said that returning to a semblance of life before the outbreak was crucial but, “It’s not Job No. 1, because right now, the house is on fire and Job No. 1 is to put the fire out.”