The United States and several Latin American countries said they would send humanitarian aid and rescue personnel to Venezuela, after two major earthquakes struck west of Caracas on Wednesday night, killing at least 32 and injuring hundreds more.

“I have instructed all agencies of our government to get ready to move quickly. We will be there for our new and great friends,” President Trump wrote in a social media post on Wednesday night.

By early Thursday, Secretary of State Marco Rubio said in a social media post that the State Department was immediately deploying search and rescue teams, medical resources and humanitarian assistance to Venezuela.

The United States was coordinating a response to the earthquake with the interim government in Caracas, according to Under Secretary of State Jeremy Lewin, the State Department’s senior official in charge of foreign assistance, humanitarian affairs and religious freedom.

President Delcy Rodríguez, who has declared a state of emergency, was installed in January after U.S. forces deposed Nicolás Maduro, the former leader of Venezuela, and took him to the United States.

Ms. Rodríguez said that 32 people were killed and 700 were injured after the two earthquakes. Each temblor had a magnitude higher than 7 and struck less than a minute apart, according to the United States Geological Survey. Videos on social media showed buildings collapsing and rescue teams combing through the rubble searching for survivors.

Spain, which once ruled Venezuela as a colony, said its foreign ministry had offered to send emergency aid to Venezuela, as did many of Venezuela’s neighbors in Latin America, including Chile, Argentina and Panama. Venezuelan authorities have requested specialized rescue and health personnel from Mexico, President Claudia Sheinbaum said.

The Dominican Republic would deploy military teams specialized in search and rescue to Venezuela early Thursday morning to help the Venezuelan authorities, said President Luis Abinader, who added that he had spoken to Ms. Rodríguez by phone to express solidarity.

Earlier, President Nayib Bukele of El Salvador said in a social media post that 300 rescue workers and paramedics were ready to depart for Caracas with equipment, medicine and supplies.

President Daniel Noboa of Ecuador said he had arranged for the immediate delivery of humanitarian aid to Venezuela, despite strained relations between the two countries. Millions of Venezuelans fleeing economic insecurity at home have crowded Ecuador’s borders, and Mr. Noboa has positioned himself as a regional ally to President Trump in countering the socialist government in Venezuela.

“Ecuador is responding with the speed and commitment that this moment demands, because despite enormous differences, humanity must always guide the actions of a leader,” he said in a social media post.

President Luiz Inácio Lula da Silva of Brazil said he had instructed the foreign ministry to assess how his country could help Venezuela.

“I reaffirm our determination to support the government of Acting President Delcy Rodríguez in the recovery of affected areas of this sister nation, whose people have given proof of great resilience in the face of adversities,” he wrote in a social media post.

Overnight, Ms. Rodríguez thanked Mr. Trump and other leaders for their offers of assistance on social media.

Prime Minister Narendra Modi of India said he was “ready to extend all possible assistance” to Venezuela.

Guo Jiakun, China’s Foreign Ministry spokesman, said in a news briefing on Thursday that China was “willing to provide assistance to Venezuela in an appropriate manner, according to its needs.”

The chef José Andrés, founder of World Central Kitchen, an international food aid group, said his Longer Tables Fund would contribute $1 million to the relief efforts in Venezuela.

Italy’s Foreign Minister Antonio Tajani said his country would ask the European Union to activate its civil protection mechanism, which coordinates the bloc’s assistance to any country experiencing a disaster.