Mr Bolsonaro had earlier refused a G7 offer of $22m (£18m) following a spat with French President Emmanuel Macron.
But following a meeting between the governors and Mr Bolsonaro, the government shifted its position on aid.
It said it would accept it as long as it had control of what to spend it on.
Why does it matter?
A record number of fires are burning in Brazil, most of them in the Amazon region. The Amazon is a vital carbon store that slows down the pace of global warming.
President Bolsonaro said last week that his government lacked the resources to fight the fires.
As international concern over the fires spread, leaders of the seven leading industrial nations meeting in France offered $22m to help fight the fires.
President Macron, who was hosting the summit, said the funds would be made available immediately – primarily to pay for more firefighting planes.
But President Bolsonaro rejected the offer arguing that the G7 countries were treating Brazil like “a colony or a no-man’s land”.
Will Brazil take the aid now?
That is not entirely clear yet. There has been a lot of back and forth on this.
After ruling out accepting the aid, President Bolsonaro softened his stance a little on Tuesday saying he would consider doing so, if President Macron apologised for insulting him by calling him a liar.
And following President Bolsonaro’s meeting with the governors late on Tuesday, presidential spokesman Rego Barros said the Brazilian government “is open to receiving financial support from organisations and countries”.
However, Mr Barros stipulated that the aid would have to have the “total governance of the Brazilian people”. There has been no response yet from the French government or the G7 countries.
The governor of Maranhão state, Flávio Dino, said he and his counterparts from other affected states had told Mr Bolsonaro that “it’s not the moment to turn down money”.
What aid has been pledged?
Apart from the $22m the G7 countries offered, Hollywood actor Leonardo DiCaprio also pledged $5m to help fight the fires.
Apple CEO Tim Cook also tweeted that the company would be “donating to help preserve the Amazon rainforest’s biodiversity” but he did not name a figure.
US President Donald Trump tweeted that Mr Bolsonaro and Brazil “have the full and complete support of the USA”
Reuters news agency says it has been told by a diplomatic source in Brasilia that the Brazilian government has accepted $12.2m from the UK government.
What is Brazil doing to stop the fires?
The government says it has deployed 44,000 soldiers to seven states to combat the fires.
Prosecutors are also investigating allegations that some of the fires were trigged by the illegal clearing of land.
The justice ministry says that federal police officers would be sent to the fire zones to assist other state agencies in combating “illegal deforestation”.
Are there more fires than in previous years?
Data published by Brazil’s space agency suggests there are. The agency, known as Inpe, says there have been more than 83,000 fires between 1 January 2019 and 27 August 2019. That is a 77% rise compared to the same period in 2018.
Nasa has also warned that the “2019 fires season has the highest fire count since 2012”.
BBC analysis has also found that the record number of fires being recorded coincides with a sharp drop in fines being handed out for environmental violations.
What are people on the ground saying?
Jorgimar Alberto, from Roraima, one of the states most affected by the fires, has seen the burning of most of the land surrounding his wooden house.
“It’s a risky situation, we have lots of crops here and everything is burning, such as the cashew trees,” he told the BBC.
“I even had to keep the animals [inside] so that they don’t burn as well. Every year it is the same, these fires are disturbing the region a lot.”
Environmental officials helped him extinguish the fires last week only for them to reignite, he said.
His wife could not stand staying in the house with the smell of burning and the threat getting ever closer, so she left.
Lumberyard owner Edson Oliveira from southern Amazonas said he did not believe the media coverage about the Amazon.
“In my opinion, the parts that are burning are the same that burn every year,” he told the BBC.
Asked what he would like politicians to do for the Amazon, he said that he would like “economic alternatives for people here, to find a way we can make use of the natural wealth we have here”.
“It’s no use pointing the finger at what’s wrong without bringing alternatives to people,” he said.