"This rainfall may cause flash flooding", the Miami-based NHC said.
At Lowdermilk Park, the beach was almost empty despite there being no signs of red tide.
The storm is expected to move away from the southwestern Florida coast and reach coastal MS and Louisiana by late Tuesday.
US oil producer Anadarko Petroleum Corp on Monday evacuated workers and shut production at two offshore oil platforms, and other companies with production and refining operations along the Gulf Coast said they were securing facilities.
In its 5 a.m. advisory, the hurricane center said the potential tropical cyclone was producing maximum sustained winds of 30 miles per hour with higher gusts and moving north-northwest at 16 miles per hour.
No coastal watches or warnings are in effect. This means the remnants of Gordon could move close to the Ark-La-Tex or possibly even into northeastern Texas by late Thursday into Friday.
Meteorologists also warn of high rip current risks due to the bad weather, advising beachgoers to swim near a lifeguard.
Police advised beach-goers to stay indoors. If unable to escape, face the shore and call or wave for help, ' the weather service in Miami said. The water could reach the following heights above ground somewhere in the indicated areas if the peak surge occurs at the time of high tide.
Parts of the northern Gulf Coast could see 3 to 6 inches rainfall Tuesday into Wednesday, from southeast Louisiana into far southern MS and far southern Alabama, according to the Weather Channel.
The NWS said in its 8:30 a.m. EDT advisory the storm was centered 20 miles west of Key Largo and 85 miles southeast of Marco Island. A storm surge watch was extended to the western Florida Panhandle. Tropical Storm warnings have been issued from the Alabama-Florida border westward to East of Morgan City, Louisiana. NWS forecasters said isolated tornadoes are also possible.