The incident happened on Kirkton Street, not far from Kirkton Primary School, shortly after the school day finished.
Emergency services have confirmed that three children and a woman in her 20s have been taken to Queen Elizabeth Hospital in Glasgow to be treated.
It is understood that the eldest child injured in the incident is 7-years-old.
The extent of their injuries are unknown at this time.
A Scottish Ambulance Service spokesperson said: “We received a call at 3.07pm to attend an incident on Kirkton Street, Carluke and dispatched 10 resources to the scene.
"One female in her twenties and three children have been transported to the Queen Elizabeth University Hospital, Glasgow.”
Inspector William Broatch, from Motherwell Road Policing Unit, said: “Around 3.10pm, police were called to James Street in Carluke, at the junction with Kirkton Street, following a report of a road crash involving a car and four pedestrians - a woman and three children.
"The pedestrians are all being conveyed to hospital for treatment. Emergency services remain at the scene and local diversions are in place.
"Anyone with information on the incident can call police on 101, quoting incident 2110 of 27 October.”
The A73 is currently closed between Glamis Avenue and Clyde Street and traffic is being diverted.
Local MSP Mairi McAllan, who represents the Clydesdale constituency, said she was "deeply worried" about the reports coming out of Carluke.
She said: "My team have been in touch with local police offering whatever help we can give.
"I want to thank the emergency services who're on the scene, and in the meantime I'm just praying like everyone else for good news that no one's seriously hurt."
Emergency services remain at the scene and local diversions are in place.