The proposed deal comes as global sales of premium beers increase at twice the rate of mainstream beers.
Grolsch has been independent for more than four centuries and has not openly welcomed the approach.
Ab Pasman, chief executive of Grolsch, said SABMiller made the offer even though "we did not look for this takeover".
However, analysts expect the high purchase price to force through a sale, resulting in the De Groen family, owners of the Dutch brewery since it was founded in 1615, agreeing to sell their 37% stake.
The move could have ramifications in the ongoing battle to buy Scottish & Newcastle, which is trying to defend itself against a takeover from Carlsberg and Heineken.
SABMiller has previously been linked with a white knight bid for Scottish & Newcastle, but Malcolm Wyman, SABMiller's chief financial officer, said the Grolsch purchase did not affect the brewer's ability to consider other deals.
Grolsch's two main markets are the Netherlands, where has it has a 15% share, and the UK, where it has a 2% share and is brewed by Molson Coors.
In addition to Grolsch, which accounts for about 90% of its portfolio, Grolsch also has other brands including Grolsch Premium Weizen and Spring Bock.
SABMiller brews Pilsner Urquell in addition to the Peroni brand, which was acquired by the company in 2003 but has failed to compete effectively with Grolsch brand in the premium beer market.