ABI has upped the £42.15 per share or £65bn it proposed offering for the South African brewer of Peroni to £43.50 a share
The Belgian brewer of Stella Artois made the revised approach to SABMiller's board on Monday.
The group did it privately in the hope of persuading SABMiller to take part in talks, Britain's Sky News cited people close to the situation as saying.
It must act before a so-called put-up-or-shut-up deadline this Wednesday.
ABI is determined to get its hands on its rival to grab more of the global beer market, where it believes growth will come from acquisitions rather than selling more beer.
But experts cautioned it against overpaying, saying constant hikes in the offer price will reduce the value of the savings from the deal.
John Colley, of Warwick Business School, said: "ABI is being forced to transfer many of the synergy benefits to SABMiller’s shareholders through constantly raising the price.
"The global beer market overall is largely flat and in some regions is declining as other beverages such as wine continue to penetrate.
"Micro-brewers and their highly differentiated cask ales also continue to make progress.
"As a consequence cost, product and distribution rationalisation become an attractive way of increasing shareholder returns. That is provided ABI does not pay too much for SABMiller."
SAB has already turned down the £42.15 approach and two previous proposals on the grounds that they undervalued the company.
US tobacco firm Altria, which owns 27% of SABMiller, has backed ABI but the Santo Domingo family, which has a 14% stake, has supported the board.
Meanwhile, Reuters reported that SABMiller's fourth largest shareholder rejected ABI's third offer as too low.
"We have confidence in the board and we will rely on their judgement. They have said the price is too low and we agree with them," the news agency quoted Dan Matjila, chief executive of Public Investment Corporation, which owns about 3% of SABMiller, as saying.
SABMiller shares fell 8.5p to 3660p in early London trading. ABI's stock dropped 0.04% to €98.26.
ABI also makes Budweiser and Corona, while Britain-based SABMiller - which holds second position - owns Peroni, Grolsch and Pilsner Urquell beers.
ABI employs about 1,000 people at its UK breweries in Magor, Samlesbury and Mortlake, as well as its company headquarters in Luton. It has at least 200 beer brands.
SABMiller has around 69,000 staff in more than 80 countries, from Australia, Zambia and Colombia to the Czech Republic, South Africa and the US, also producing more than 200 beers.
Formerly known as South African Breweries, SAB acquired US rival Miller Brewing Company in 2002, resulting in the renaming of the group as SABMiller.
A merged group would be worth about £177bn and would add AB InBev's dominance of Latin America to SABMiller's strong presence in Africa, as well as their breweries in Asia.