A search has been launched for the two children missing from their home in Irvine since early August.
Willie Boy O'Neill, 3, and his brother Tojo O'Neill, 2, are believed to be with their parents Claire Watson, 32, and Robert O'Neill, 39.
North Ayrshire Council staff had been in touch with the couple recently over concerns for the childrens' welfare, according to police.
The family, formerly of Caldon Road, were last seen at Rivergate Shopping Centre in Irvine shortly before 3pm on Sunday August 3.
Willie Boy is described as being of slim build with fair hair, while Tojo is stocky with fair hair.
Police said officers have made "considerable efforts" to track down the family.
They have made "extensive enquiries" in England, where there are family connections, according to a police statement.
Police Scotland, local area commander Chief Inspector Martyn Hopkins, said: "I would appeal to their parents or to anyone who knows where they are to contact Irvine Police Office via non emergency number 101 or North Ayrshire Council as a matter of urgency."
A spokeswoman for North Ayrshire Council indicated support for the police appeal but declined to comment on the circumstances surrounding their disappearance.
She said: "Anyone with information on the whereabouts of the two children should contact Irvine Police Office on 101 or North Ayrshire Council on 01294 311505 as a matter of urgency."