Police have shot a woman who allegedly ran at officers with a kitchen knife in a park in Melbourne's inner north.

The woman, in her 50s, was taken to hospital with serious injuries to her thigh following the shooting at Penders Park in Thornbury.

Police received reports of a woman carrying a knife near Collins Street about 2:50pm.

It is believed that police challenged the woman to drop the weapon, before deploying capsicum spray and firing one shot just before 3pm.

She was arrested and taken to hospital.

Armed Crime detectives will undertake an investigation into the incident, which will be overseen by Professional Standards Command. 

Anyone who witnessed the incident or has footage has been urged to contact Crime Stoppers on 1800 333 000.