The storm is expected to turn northwest toward the Yucatan peninsula and into the western Gulf by early next week.
The long-range track shows what could be Tropical Storm Marco approaching the north-central U.S. Gulf Coast on Tuesday.
The Hurricane Center said Tropical Storm Genevieve had maximum sustained winds of 45 mph (75 kph) and was centred about 155 miles (255 kilometres) west of Cabo San Lazaro, Mexico. The storm may eventually pose a threat to the Texas or Louisiana coasts by next Tuesday. As of now, the Cape Fear Region is not in the official National Hurricane Center forecast track cone for Laura, but please stay alert in the unlikely event that that changes.
"Laura could bring storm surge, rainfall, and wind impacts to portions of Cuba, the Bahamas and Florida early next week", notes the NHC, adding that the northeastern US Gulf Coast could be affected by the storm by the middle of next week.
Forecasters from the National Hurricane Center are closely monitoring the systems.
A hurricane watch is in effect for Punta Herrero to Cancun, Mexico.
The official forecast map doesn't show it becoming a hurricane, but forecasters said the depression could be near or at hurricane strength (winds of 74 miles per hour or higher) when it nears the Yucatan Peninsula on Saturday.
Residents in areas of concern are urged to monitor the forecast closely and prepare in advance in case of evacuations.
En route, it's likely to soak flood-prone eastern Honduras, the Cayman Islands and parts of the Yucatan.
NHC forecasters say that this motion is expected to continue through this evening.
This wave is expected to move over the far eastern tropical Atlantic on Friday, with some slow development possible through the weekend as it moves west-northwestward at 24-32 km/h across the eastern tropical Atlantic.
It is too early to know exactly how strong the systems will become or where they will impact.
"Hurricane forecasting is a delicate act".
Tropical Depression Thirteen is forecast to become Tropical Storm Laura later today.