The stand-off helped Scotland to victories over Italy and Australia and was then selected as one of the 'Geography Six' for the British and Irish Lions series.
The summer signng from Bath was carried off early in the second half in with a cut to his right thigh and taken to hospital in Galway for treatment.
Hastings, who made his Glasgow debut in the pre-season game against Northampton at Bridgehaugh, had replaced centre Sam Johnson, who took a head knock in the 18-10 victory and Warriors forwards coach Jonathan Humphreys said yesterday: "Sam is undergoing HIA [head injury assessment] protocols. Adam is going to be out for a couple of weeks with a nasty gash in his leg. He had a fair few stitches.
"The big thing for him is that he does not get it infected. It should be three weeks or so."
"Seymour is still a couple of weeks away but Finn will be available soon," said Humphreys.
"There is a lot of competition in certain places. It is about giving people opportunities because before we know it a large part of this group is away to play for Scotland. It is a big thing that this squad is a lot more competitive when everybody is away.
"You will see changes over the next three or four games. It is important we do that. You cannot win this competition with just a small group of players. It is based on 40-odd guys."