Bain, who arrived from Alloa Athletic, was left behind in Hungary to undergo surgery when the rest of his team-mates returned from their pre-season training camp.
Dundee-born Adam, who started his career with brother Charlie at Rangers, joins from Airdrie and has also played for Cowdenbeath and St Mirren.
Last season's first-choice, Kyle Letheren, has also been injured but should shake off a calf injury in the coming days.
"We needed to bring someone in," said Paul Hartley, the Dundee manager. "Kyle's had a knock, and hasn't done a lot, so Grant could be in line to play on Sunday [against Manchester City in a friendly], which is fantastic for him."