The 25-year-old victim was on Wellington Street about 12.15am today when he was attacked.
The man was rushed to Inverclyde Royal Hospital and is said to be in a serious, but stable condition.
The women were both uninjured, but have been left extremely distressed by what happened, police said.
The incident is being treated as attempted murder.
The suspect is white, around 6ft tall, with a slim build.
He was wearing a black hooded top and black tracksuit trousers and ran off towards Greenock Health Centre after the attack.
Detectives are carrying out door-to-door enquiries and checking CCTV footage from the area.
Detective Inspector David Wagstaff said: “Our enquiries are at an early stage but from what we know so far, we believe the victim was the intended target.
“The motive, however is unknown and we are in the process of trying to establish exactly why this attack has taken place.
“I am appealing to members of the local community to contact us with any information they may have about this incident or any knowledge of who the suspect is. Please get in touch with us.”