Garcia on Monday demanded a detailed explanation from Chile after Peru detained one of its own air force officers on suspicion of treason for allegedly spying for Chile.
Chile's government has denied any involvement, and Garcia on Monday stopped short of suggesting the spat would hurt bilateral trade worth more than $3 billion a year.
"The expressions we heard yesterday, which I would call offensive and haughty, do not contribute to the integration and cooperation that neighbors should enjoy," Bachelet said in remarks at the opening of an air traffic control facility in Santiago.
"I think this is a time, if we really want to work for the welfare of our peoples, for respect above all, as well as responsibility from the authorities," she added.
Also on Tuesday, Peru said its investigation into the spy case had widened and uncovered evidence suggesting at least six other people were involved, besides the air force officer initially identified.
Garcia's cabinet head declined to give further details on the suspects.
Peru cited the alleged espionage as its reason for quitting the Asia-Pacific Economic Cooperation summit in Singapore this past weekend. Garcia was scheduled to meet with Bachelet at the summit.
Chilean officials have suggested he timed the revelation to create a scandal at the summit, where leaders were holding talks on regional integration.
Peru and Chile have frequently sparred over their border since Chile won the 1879-1883 War of the Pacific and took over a slice of mineral-rich land from its neighbor.
Last year, Peru filed suit against Chile at the International Court of Justice in The Hague, disputing its maritime territorial rights and demanding more control of the rich Pacific Ocean fishing waters between them.
(Reporting by Bianca Frigiani in Santiago and Dana Ford in Lima, writing by Simon Gardner, editing by Vicki Allen and Todd Eastham)