"Italy will present a proposal in a transparent way, and I think that we will find a sensible solution in Europe," Merkel said.
Italy's request comes after a devastating earthquake struck four of its central regions last week, killing 293 people and causing millions in damage.
Renzi said that Italy "will take what it needs" for his proposed, far-reaching Home Italy project to make the country's buildings anti-seismic.
"We will follow EU rules" on budget flexibility and "we will take everything we need and that is necessary, but it is key that we start off by spending money properly," Renzi said.
Merkel added that Germany will finance the rebuilding of a school in the earthquake-stricken central Italy.
"Our government will make a contribution, as will German entrepreneurs and the German football league," she said.
Earlier, Renzi had announced that Ferrari carmaker will auction off one of its luxury models for earthquake relief.
"The starting price will be one million euros," Renzi said in Maranello, the hometown of Ferrari where the Italy-Germany summit was held.
He also thanked Ferrari President John Elkann and the CEO of parent company Fiat Chrysler Automobiles, Sergio Marchionne.
Also at the summit, Renzi announced that former Emilia Romagna governor Vasco Errani will be commissioner for post-quake reconstruction.
"He'll be nominated tomorrow," Renzi said.
"He's been governor of Emilia Romagna, which is still standing after a powerful quake four years ago. We pick the same team."
Italy "needs to spend the money it already has in the right way," Renzi said.
"There will be total transparency... and absolute legality."
Renzi and Merkel had earlier met the fire fighters who rescued an eight-year-old girl in the village of Pescara del Tronto, which was hard-hit by the quake.
The meeting in the Modena-area town of Maranello, home to Ferrari luxury carmaker and its Formula One racing team, was attended by delegations from the Civil Protection Department, the fire department, and rescue dog Leo, who first detected the child trapped under meters of rubble.
Giorgia was extracted from the debris of her home 16 hours after the magnitude 6.2 quake struck last Wednesday. Her 10-year-old sister was found dead beside her while her parents were both pulled out alive.
ANSA