The council's building control team has stepped in to deal with 202 Millroad Drive, in Calton, indicating that demolition is looming.
It came as a Scottish Water investigation concluded its apparatus had not caused the subsidence.
As the Evening Times has reported, two families in total have so far been evacuated from the street.
Neighbours have also raised safety fears over the state of the end terrace house, which is being swallowed up by a 10ft hole.
There are concerns that the whole street is affected by the severe subsidence because cracks are appearing in other homes.
A city council spokesman said: "The council's building control team intends to serve a dangerous buildings notice on the property, and are in discussions with the owners on their plans."
Thenue Housing Association, which owns some of the properties in the row of housing, has been in talks with their tenants.
A spokesman said: "On Wednesday, the housing association was made aware that Glasgow City Council intends to serve a dangerous buildings notice on the owner-occupied property at 202 MIllroad Drive.
"This follows an assessment of the property by the city council's building control department.
"It is anticipated that this notice may result in the demolition of the property.
"Thenue Housing continues to monitor this situation on a daily basis in consultation with our structural engineering advisors."
The cause of the ground movement remains a mystery and talks on how to go forward are continuing.
It is understood the notice will be served after final examinations around the property.
A spokesman for Scottish Water said: "We have completed our investigation and our specialist engineers are of the view that our apparatus has not caused this subsidence.
"We have outlined our position to the relevant organisations."
The alarm was first raised in December and the couple were forced to flee their home just before Christmas.
Just weeks later, Terence Laird, 75, who had lived in the home attached to the sinking property for 32 years, was moved out by Thenue.
On Monday, we told how another Thenue tenant Isabel Weir, 52, feared for her own safety and that of her 20-year-old daughter after she noticed cracks in her own home.
Mrs Weir, who lived next to Mr Laird and two doors down from the collapsing home, said the head of housing had visited her to provide reassurance.
She said: "I've been told we're safe here for now.
"I've said I will move out if they have to demolish the house but I would want to move back in."
The Thenue spokesman said: "At all times the safety of our tenants is our prime concern."