Feltham Police Station officers were alerted by commotion outside the front of the station, in Hanworth Road, at around 3am on Friday (August 18).
On-duty officers went outside and found a man, aged in his 30s, suffering from stab injuries.
London Ambulance Service attended and took the victim to a west London hospital with possible life-threatening injuries, the police said.
It is believed the victim was stabbed in Florence Road, which is just around the corner from the police station.
A Metropolitan Police spokesman said: "Officers and the Dog Unit attended and searched the area.
"A 24-year-old man was arrested at an address in Feltham on suspicion of grievous bodily harm with intent.
"He has been taken to a west London police station where he remains in custody.
Roads in the area were closed, and reopened around 8am.
Anyone with information is asked to contact police on 101 or Tweet @MetCC.
Information can also be reported to Crimestoppers on 0800 555 111 or online at crimestoppers-uk.org.
Keep up to date with the latest news in west London via the free getwestlondon app.
You can even set it to receive push notifications for all the breaking news in your area.