Venezuelan Foreign Minister Elias Jaua described the diversion by saying: “This is an attack against President Morales’s life.”
“We express our solidarity with Evo and the brave Bolivian people. Our America cannot tolerate so much abuse,” Ecuadorean President Rafael Correa added on Twitter.
In La Paz, the presidential palace said protest rallies were already planned outside the embassies of the United States, France, Portugal and Italy.
Snowden has remained quiet and out of sight of reporters since arriving at Sheremetyevo Airport from Hong Kong.
Late Monday, he accused Washington of pressuring foreign leaders to refuse him refuge.
Snowden’s latest major leak about US spying on EU countries has angered many European governments and threatened to derail preparations for delicate talks on a huge free trade deal between Washington and Brussels.
On Wednesday, Austrian Interior Minister Johanna Mikl-Leitner said Europe’s trust in the United States had been “shattered” by the allegations.
France meanwhile called for a temporary suspension of the trade talks.
“This is not about stopping negotiations on the free trade agreement, but it does seem wise to temporarily suspend them, probably for a period of 15 days, to avoid controversy and to give time to obtain the requested information,” Najat Vallaud-Belkacem said after a cabinet session.
EU ambassadors are due to discuss the US spy allegations in Brussels on Thursday.