Claudette "has swerved northeast and will intensify to hurricane status in the next few hours," Mexico's National Meteorological Service said in a statement.
"It will endanger the coast of Quintana Roo," said Carmen Segura, national civil defense coordinator. The government began late Thursday to evacuate Quintana Roo of 1,500 residents.
However, residents there have remained calm, declining to stock up on foodstuffs or taking other preventive measures such as sealing windows.
US weather forecasters issued a tropical storm warning for Claudette, which could become the first hurricane of the season as it cuts across the Caribbean.
At 2100 GMT, Claudette was centered about 340 kilometers (210 miles) southeast of Cozumel, Mexico, off the northeastern tip of the Yucatan. The storm was advancing northwest at 22 kilometers (14 miles) per hour.
Claudette, packing winds of nearly 110 kilometers (70 miles) an hour with gusts of up to 140 kilometers (87 miles) per hour, was expected to reach Mexico's Yucatan peninsula late Thursday or early Friday.
Mexico's emergency covered the state of Quintana Roo, along the peninsula's Caribbean shore.
Mexican meteorologists predicted that by early Friday, Claudette would be 140 kilometers (90 miles) from Quintana Roo, with winds reaching 120 kilometers (100 miles) per hour and gusts of 150 kilometers (93 miles) per hour -- a Class One hurricane.
Mexican authorities issued a storm warning between Chetumal and Catoche on the Yucatan peninsula.
Tropical storm warnings for Grand Cayman island were to be downgraded late Thursday, the US National Hurricane Center said.
After buffeting the peninsula, Claudette will cross the Gulf of Mexico and head for Texas, with landfall expected early next week, the US hurricane center said.
The Atlantic hurricane season runs from June to November. Experts have predicted more storms than usual this year.