Russell, who was transferred from University Hospital in Galway to Beaumont Hospital in Dublin, is expected to be discharged later this week.
His return to training date will be determined when he is back in Glasgow, but he is set to miss Scotland's two-Test tour to Japan this summer.
Glasgow Warriors would like to thank the Connacht medical team, players and staff for their support. The club would also like to thank the University Hospital in Galway and Beaumont Hospital in Dublin for all their help.
Finn and his family really appreciate all the messages of support, with which they have been inundated.
Other players are still being assessed by the Glasgow Warriors medical team following Saturday's match against Connacht.
Further updates will be provided when available.
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