The Warriors meanwhile need just two points to secure their presence in the play-offs which could either be earned via a draw or by gaining two bonus points.
But Weir, who was this week named in the RaboDirect PRO12 Dream Team, will be going for the win against the Irishmen with the knowledge that a strong performance will further his case for a place in Scotland head coach Andy Robinson's touring party to Australasia this summer.
"Connacht is going to be tough," Weir told the Scotsman.
"Watching the football this week you see Chelsea losing to Newcastle and people who haven't watched Connacht maybe don't realise how well they're playing.
"They're not blessed with the big names of Leinster or Munster, but they have a great team spirit and have had some cracking results in the league and in the Heineken Cup.
"Obviously, the (Scotland) summer tour is on the back-burner but playing well for Glasgow is the main focus.
"If I can do that then I would be in a position to put my hand up for selection and if we can win this weekend and get into the play-offs and do well there, which I know we're capable of doing when we play the way we can, then we will all be able to turn some heads."
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