BOGOTA, Colombia—President Juan Manuel Santos said Wednesday he’s willing to meet with his Venezuelan counterpart to resolve a border crisis that has led more than 20,000 undocumented Colombian immigrants to flee Venezuela and fueled a bitter row between the two countries.

Just hours before, in a four-hour speech on his state television program, Venezuelan President Nicolas Maduro said he was closing the main border crossing between the Venezuelan state of Apure and Colombia’s Arauca. Mr. Maduro had already closed two other major border crossings in the Venezuelan states of Tachira and Zulia, paralyzing once-thriving legal and illicit commerce between the two countries.

Mr. Santos said in televised remarks from the presidential palace that a solution cannot be reached “expanding the closings” of the border. “Expanding the closings is not going to bring us closer to a solution to the problems on our border,” said Mr. Santos. “What’s required is an exchange that’s respectful, frank, sincere, based on facts and realities.”

Mr. Maduro, in his remarks on Venezuelan state television, said that as part of his anti-smuggling campaign he was also expanding “a state of exception” to 10 Venezuelan communities in largely rural Apure state and to the more industrial Zulia to the far northwest. The emergency measures have already been in place in 14 towns, giving authorities free range for warrantless searches and seizures and the banning of unauthorized protests.

The Venezuelan leader also said his country was “fully disposed” to a face-to-face meeting with Mr. Santos.