Cops were called to the incident at Glen Lee at around 12.10pm on Monday, November 18.
A 26-year-old man has since been arrested in connection and the force has confirmed that an investigation has been launched.
A police presence remains at the scene on Tuesday morning, with a perimeter around the flats.
One police van was parked nearby and an officer could be seen standing inside the communal entrance to the flats.
The force previously said that officers were still at the scene but assured there was no risk to the public.
A Police Scotland spokesperson said: “Around 12.10pm on Monday, November 18, 2024, we were called to the death of a woman at a flat on Glen Lee in East Kilbride.
“A 26-year-old man has been arrested in connection with the incident and enquiries are ongoing.
“Officers remain at the scene however there is no wider risk to the public.”