The two countries have fought three wars against each other since independence from Britain in 1947. Despite a peace process that began in 2004, tensions remain high, and each country wants to avoid showing any weakness to the other.
India has called on Pakistan to crack down on militant groups operating out of Pakistan, particularly Lashkar-e-Taiba, which has been blamed for the Mumbai attacks that left more than 160 people dead.
Pakistan has carried out raids on a charity believed to be linked to Lashkar, but also urged India to provide further evidence.
Abdullah Ghaznavi, Lashkar's chief spokesman, denied involvement in the Mumbai attacks, saying his group only targets Indian forces and Indian defense installations as part of efforts to force India out of its portion of the disputed Kashmir region.
"This is a jihad, and it will continue," Ghaznavi told The Associated Press in a call Sunday from an undisclosed location.
He also claimed his group has "no direct or indirect links" with the Taliban or al-Qaida.
"We neither finance them nor support them," he said.
Pakistani officials said Indian aircraft entered one to three miles (two to four kilometers) inside Pakistan's section of Kashmir and over the eastern city of Lahore on Saturday.
Pakistani jets chased the Indian aircraft back over the border, authorities here said.
Both sides are usually careful to avoid such territorial violations, and it was unclear how two separate but apparently accidental incursions could occur on the same day.
Zardari tried to dismiss the incidents, calling them "technical incursions" that had been blown out of proportion.
"Incursions do happen," he said at a news conference with British Prime Minister Gordon Brown, saying the planes were flying at about 40,000 feet (12,000 meters) when they executed a turn that "slightly entered Pakistan soil."
India denied its aircraft crossed into Pakistani airspace.
"There has not been any airspace violation by the Indian air force as has been alleged," Indian air force spokesman Mahesh Upasani said Sunday.
Pakistan Information Minister Sherry Rehman said in a statement that the Indian Air Force had told Pakistan the incursion was "inadvertent."
"There is no need for undue alarm," she said.
Lashkar-e-Taiba and the charity allegedly linked to it have a strong presence in both areas over where the Indian planes were alleged to have flown.
Pakistan has denied any official link to the Mumbai assault but has arrested some alleged plotters while demanding India hand over evidence to aid in their prosecution.
While Pakistan has offered to help in India's investigation, India has been reticent to share the information it has gleaned so far, finding itself in the awkward position of having to investigate terrorist attacks hand-in-hand with its longtime nemesis.
Zardari said he saw a joint probe as a chance to build some diplomatic bridges.
"We consider this an opportunity to cooperate with India, to take the relationship with India to another level," he said.
Singh said India hopes relations can be "normalized."
"This is my belief that all issues can be resolved through mutual wisdom and cooperation," he said, while making clear that New Delhi's tolerance has limits.
"Our good intentions should not be misconstrued as our weakness," he said.