Venezuela’s aid predicament has mirrored the international alliances forged by the competing sides. The United States and the European Union nations have become the main donors for humanitarian projects launched by Mr. Guaidó. China and Russia have delivered relief supplies to Mr. Maduro.

Mr. Guaidó has opened three aid depots on Venezuela’s borders in Colombia, Brazil and Curacao, which have collected at least $100 million worth of supplies donated by the United States Agency for International Development and other allies. Mr. Maduro called the USAID supplies a “Trojan horse” aimed at toppling his government and has vowed to prevent it from entering the country. His vice president, Delcy Rodriguez, even called the opposition’s supplies “cancerous.”

Mr. Guaidó’s plan to bring basic supplies into Venezuela in February, intended to help the population but also to bolster his standing and discredit Mr. Maduro, was stymied by Mr. Maduro’s armed forces and paramilitaries. At least seven people were killed and dozens were seriously injured during the chaotic and inconclusive standoff along the country’s border.

The Red Cross said its relief in Venezuela will not be linked to any political parties.

To maintain impartiality, the Red Cross said supplies such as surgical kits and basic medication will initially be delivered directly to the eight hospitals that the organization owns in Venezuela, bypassing the government’s distribution networks. The organization also aims to equip all Venezuelan hospitals with power generators to reduce the number of deaths caused by the rolling blackouts affecting the country.

Mr. Rocca said the organizations would also consider distributing medical donations collected by the opposition in Cúcuta, Colombia, if they met their standards.

The Red Cross was granted the permission after personal meetings with Mr. Maduro and Mr. Guaidó, said a person involved in negotiations, who spoke on condition of anonymity because he was not authorized to discuss confidential talks.

The announcement comes days after the United Nations made a confidential plea to the country’s two rival leaders to end their political dispute over aid, pointing out that nearly all of the country’s population of 32 million has been thrust into poverty.