The couple from Newcastle were part of a holiday coach group who were visiting Scotland as tourists.
It is understood they were attempting to cross a road on Queen Street when they became trapped under a double decker bus. A partition was erected around the front of the bus as firefighters worked to free the couple.
The man, 76, and his wife, 72, were taken to Glasgow Royal Infirmary where their condition improved from critical to serious.
Police are investigating the accident with the cause as yet to be determined.
A source close to the investigation told The Herald Scotland: “The man and his wife will remain in hospital for some considerable time.
“Both have significant life-changing injuries and required limb surgery, which has taken place.
“They will need a lot of support from their family.”
Sergeant Andrew Mair added: “Where the crash happened is an extremely busy shopping area.
“I would appeal to any witnesses who have not yet spoken to police to contact us by calling 101.”