The authorities in Lima sent a formal letter of protest to Caracas.
The dispute started after Venezuela's foreign minister called Peru's president a coward and a dog, obedient to the United States.
Peru's Foreign Minister Ricardo Luna said his country would not tolerate insults.
During a recent trip to the US, Peruvian President Pedro Pablo Kuczynski said Washington did not invest much time in Latin America as it was like a well-behaved dog.
But he said Venezuela was a big problem.
The remark irked the Venezuelan President Nicolas Maduro, who called for an apology over the weekend.
On Monday, his Foreign Minister, Delcy Rodriguez, said Mr Kuczynski was the only dog, always wagging its tail for the American empire.
Mr Luna said Peru's president was using a metaphor. He said he meant Latin America was not as controversial as other regions.
Since centre-right governments came to power in Brazil and Argentina in recent months, President Maduro has been increasingly isolated in Latin America.