Tropical Storm Fay moved over the western end of Cuba on Monday morning on its way toward the eastern Gulf of Mexico and Florida Keys, and could become a hurricane before making landfall in Florida tomorrow.

Officials at the National Hurricane Center said Fay could increase to near-hurricane strength by Monday evening when it should reach the Florida Keys. It’s due to make landfall on the Florida peninsula sometime Tuesday.

The Miami Herald reports that Fay’s outer bands began to reach the Keys and South Florida early Monday morning, and new storm warnings were posted in Miami-Dade and Broward counties. Miami-Dade and Broward counties were put under a tropical storm warning, and the Keys on a hurricane watch. Those three areas are also under flood watch. The storm could dump four to eight inches of rain before heading up Florida’s Gulf Coast.

Traffic getting out of the Florida Keys was brutal at times on Sunday as officials urged the evacuation of thousands from the path of the approaching storm, CNN reported. Key West Mayor Morgan McPherson said as many as 25,000 people could be leaving.

All tourists were urged to evacuate but many bars and restaurants remained open.

At 5 AM ET, Fay, with maximum sustained winds of near 50 mph with higher gusts, was heading north-northwest at about 12 mph. The storm was to blame for the deaths of at least five people in Haiti and the Dominican Republic over the weekend.