Weir began his career with Glasgow and played over 100 games before leaving for Edinburgh in 2016.
He signed for Worcester in 2018 and has played a major in helping maintain their English Premiership status.
No deal has been signed with Glasgow, but BBC Sport claims talks are progressing well and both parties are hopeful of reaching an agreement.
Weir still has the option of staying at Worcester, with one other Premiership club also expressing an interest.
Glasgow's move for Weir comes after Warriors and Scotland fly-half Adam Hastings this week announced he is to leave the club to join Gloucester next season.
In other news, Huw Jones has been linked with a potential move to London Irish next season.