Cheryl McKinlay, 26, and her son Kieran Leggate, five, were last seen at their home in William Drive, Hamilton, South Lanarkshire on Wednesday morning.
It is thought they may have travelled to Blackpool in England.
Police Scotland appealed for information about their whereabouts.
Chief inspector Louise Skelton said: "Cheryl, who also uses the names Emma Yates, Siobhan McCallum and Cheryl Leggate, and her young son are believed to have travelled to Blackpool and were in the Edward Street area earlier this morning.
"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."
She is described as 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.
Kieran has short brown hair and blue eyes.