(Updates with Katia weakening, details on rain threat)
MEXICO CITY, Sept 9 (Reuters) - Storm Katia weakened to a tropical depression on Saturday as it moved into the interior of Mexico, but it could still dump heavy rains on areas that have absorbed large amounts of precipitation and been shaken by a massive earthquake in recent days.
The U.S. National Hurricane Center said that as a depression, Katia was blowing maximum sustained winds of nearly 35 miles (56 km) per hour and should dissipate over the mountains of central eastern Mexico later on Saturday.
Mexico is dealing with the aftermath of a huge quake that struck on Thursday night, and President Enrique Pena Nieto said on Friday that Katia could be especially dangerous in hillsides rocked by the magnitude 8.1 tremor.
The earthquake, the strongest to strike Mexico in more than 80 years, killed at least 61 people.
Katia, which brought rain to the state of Veracruz when hitting the coast late on Friday, was about 115 miles (185 km) west northwest of the Gulf Coast port of Veracruz early on Saturday morning, the NHC said.
Officials in Veracruz say Katia could cause landslides and flooding. They urged people living below hills and slopes to be ready to evacuate.
Mexico's national emergency services said this week that Katia was worrying because it is very slow-moving and could dump a lot of rain on areas that have been saturated in recent weeks.
State energy company Pemex has installations in and around the coast of Veracruz but has not reported any disruption to its operations there.
As Katia reached the Mexican Gulf Coast, Hurricane Irma, one of the most powerful Atlantic storms in a century, walloped Cuba's northern coast.
Millions of Florida residents were ordered to evacuate after the storm killed 21 people in the eastern Caribbean and left catastrophic destruction in its wake.
Meanwhile, Hurricane Jose continued to move northwestward in the Atlantic and was blowing winds of 145 mph as a Category 4 storm about 160 miles east of the Northern Leeward Islands early on Saturday morning. (Writing by Dave Graham; Editing by Lisa Von Ahn)