Cheryl McKinlay, 26, who uses a series of aliases, was last seen on Wednesday morning at her home in William Drive, Hamilton in South Lanarkshire, with son Kieran Leggate. She is understood to have recently been dealing with personal issues and police said it was vital that she and her son were found.
Officers believe the missing pair have travelled to Blackpool in England after receiving information that they were seen in the coastal Lancashire seaside town yesterday.
Chief Inspector Louise Skelton, of Police Scotland, said: "Cheryl has been dealing with some personal issues recently and she may be suffering from poor health at the moment, so it is imperative that she and her son are found as soon as possible.
"I would ask that she contact her family or police to let them know that she is safe and well."
Ms McKinlay is also known to use the names Emma Yates, Siobhan McCallum and Cheryl Leggate. Police said that they believed she had been in the Edward Street area of Blackpool yesterday morning.
Police officers issued photographs of the pair as they appealed for help in tracking them down. Ms McKinlay is described as being of medium build, approximately 5ft 10in tall, with short brown hair.
She has a large scar on the top of her left arm, at the shoulder, and wears glasses. It is not known what she was wearing when she disappeared.
Kieran has short brown hair and blue eyes.
Anyone who has information about her whereabouts is asked to contact officers at Hamilton Police Office on the Police Scotland non-emergency number 101.