A woman was found dead at the scene.
The force has arrested a 26-year-old man in connection with the death and an investigation is underway.
Police remained at the scene on Monday but there is no wider 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.”
