Friday's opening of the summit will mark the first face-to-face encounters between Alvaro Uribe, the president of Colombia, Chavez and Correa since the diplomatic row began.
Chavez predicted the summit would be "positive."
"People should go cool off a bit, chill out their nerves," he said at his hotel on Friday before leaving for the summit before the meeting.
"It is going to help the debate. We have to debate, talk, and this is the first step toward finding the road."
Arriving for the summit on Thursday, Correa said he also hoped to put an end to the row at the summit.
"It's as easy as hearing unconditional apologies from [Colombian] President Alvaro Uribe, without fallacies," Correa said.
Argentina, Guatemala and Mexico were among those coming to the Rio Group summit that was planned long before Colombia sparked a crisis with the raid last week in Ecuadorean territory to kill more than 20 Revolutionary Armed Forces of Colombia (Farc) fighters, including senior leader Raul Reyes.
Felipe Calderon, Mexico's president, said he hopes that "we can have a dialogue in a sensible way to find a peaceful solution" at the summit, but added that he doubted one would be found quickly.
Correa, who visited six countries to lobby against Uribe in a week full of vehement rhetoric on both sides, said Colombia must pledge not to violate its neighbours' territory any more and cease accusations they are supporting the Farc.
Uribe, who is popular at home for military victories against the Farc, did not talk about the crisis to reporters on arrival in Santo Domingo, the Dominican capital.
But Uribe says he has already apologised and Colombia demands more co-operation from its neighbours in fighting the Farc, who have killed and kidnapped thousands of people in a four-decade conflict.
The Colombian president also refused to rule out future military incursions into Ecuador or Venezuela, saying he first needs assurances from Correa and Chavez that they are not harbouring rebels.
Ecuador made a sign it would move in that direction when, on the eve of the summit, it announced it had captured five Colombian rebels in an Amazonian region near its shared border with Colombia.
"In an operation by the armed forces, five presumed guerrillas were found. Farc guerrillas," Gustavo Larrea, the security minister, said.
Chavez has led the region in general condemnation of Colombia's violation of Ecuador's sovereignty and he says the United States is militarising Colombia in order to attack Venezuela.
Nicaragua joined Venezuela and Ecuador on Thursday in cutting off diplomatic relations with Colombia.
The US has backed Colombia, its closest South American ally and recipient of billions of dollars in US aid for fighting rebels and the cocaine trade.