French President Emmanuel Macron said on Sunday that a deal to provide "technical and financial help" was close, the BBC reported.
He said the leaders have "all agreed on helping those countries which have been hit by the fires as fast as possible.
"Our teams are making contact with all the Amazon countries so we can finalise some very concrete commitments involving technical resources and funding," the French President added.
Macron's remarks come a week after he described the fires as an "international crisis" and pushed for them to be prioritised at the G7 Summit.
UK Prime Minister Boris Johnson said that Britain would provide 10 million pounds to protect the Amazon rainforest.
After facing mounting pressure, Brazilian President Bolsonaro on August 23 authorised the military to help tackle the blazes.
Defence Ministry officials said on Sunday that military intervention has been authorised in seven states, adding that warplanes were also drafted in to dump water on the affected areas.
Bolsonaro tweeted on Sunday that he had also accepted an offer of support from Israeli Prime Minister Benjamin Netanyahu, the BBC reported.
Meanwhile, neighbouring Bolivia was also struggling to contain fires burning in its forests.
On Sunday, President Evo Morales suspended his re-election campaign and said he was prepared to accept international help to tackle blazes in his country's Chiquitania region.
Political leaders, celebrities and environmentalists have called for action to protect the Amazon, known as the "lungs of the world" for its role in absorbing carbon dioxide and producing oxygen.
The rainforest is also home to about three million species of plants and animals and one million indigenous people.
Thousands of protesters have also taken to the streets across the world calling on governments to intervene.
On Sunday, Pope Francis also joined the call to protect the rainforest.
"We are all worried about the vast fires that have developed in the Amazon. Let us pray so that with the commitment of all, they can be put out soon. That lung of forests is vital for our planet," he told thousands of people in St Peter's Square.
( With inputs from IANS )