The agency quoted sources close to the deal, and said the negotiations continue, with the final figures in the mega-deal apt to change.
Monsanto's board of directors met Tuesday evening. Several news outlets including the German newspaper Rheinische Post said Bayer is offering an increased $129 (115 euros) per share for Monsanto, compared with the $127.50 it proposed last week.
The new offer would value Monsanto at $56.5 billion.
Bayer made a first all-cash offer in May of $122 per share, which Monsanto rejected as "incomplete and financially inadequate". The German company's second offer was $125 per share.
A merger of the two companies would create a new global leader in genetically modified seeds and pesticides.
As part of its latest proposal, Bayer is also offering to double, to $3 billion, the compensation it would have to pay to Monsanto if the deal is stopped by competition regulators, Rheinische Post said.
The proposed tie-up has rung alarms bells for some farmers who fear the power of the combined company in the market for seeds and pesticides, while opponents of genetically-modified food in Europe worry about Monsanto's influence on the continent.
- AFP