Bolsonaro had earlier refused a G7 offer of $22 million following a spat with French President Emmanuel Macron. Offering aid, the latter had said that the fires were "two times the surface area of France" and damaging the "most important lungs of the planet".
The Brazilian leader, however, refused the aid and said his country was not a "colonial territory" and that there was "a hidden agenda" behind Macron's offer.
But following a meeting between the Governors and 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.
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.
After ruling out accepting the aid, Bolsonaro softened his stance a little on Tuesday saying he would consider doing so, if Macron apologized for insulting him by calling him a liar.
And following 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", the BBC reported.
However, Barros said 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.
Maranhao state Governor Flavio Dino said he and his counterparts from other affected states had told the Brazilian President that "it's not the moment to turn down money".
Apart from the G7 aid, Hollywood actor Leonardo DiCaprio also pledged $5 million 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 Bolsonaro and Brazil "have the full and complete support of the USA".
The government says it has deployed 44,000 soldiers to seven states to combat the fires.
Prosecutors were also investigating allegations that some of the fires were trigged by the illegal clearing of land. Data published by Brazil's space agency Inpe said there have been over 83,000 fires between January 1 and August 27. That is a 77 per cent rise compared to the same period in 2018.
Nasa has also warned that the "2019 fires season has the highest fire count since 2012".
( With inputs from IANS )