The second-biggest brewer in the world, SABMiller, is coming under pressure over a takeover sought by its larger rival, even as it spurned the most recent approach as one that “still very substantially undervalues” the company.

On Wednesday, its suitor, Anheuser-Busch InBev, said that it was now willing to pay about $104 billion for SABMiller.

By publicly airing its latest informal offer, Anheuser-Busch InBev is hoping to delicately maneuver its longtime rival into what would be the biggest beer merger in history. A deal would give the maker of Budweiser and Stella Artois important footholds in Africa and Latin America, two big markets where the growth of major beer brands has proved strong.

The move comes as a deadline for the deal looms. Under British takeover rules, Anheuser-Busch InBev must make a formal offer by Oct. 14 or walk away for six months, in what the deal-making industry calls the “put-up-or-shut-up” mandate. If the beer giant wants to avoid waging a potentially costly hostile bid or dropping its pursuit for now, it must entice its reluctant rival to enter merger talks by then.