Willie Boy O'Neill, three, and Tojo O'Neill, two, formerly of Caldon Road, were last seen on 3 August.
They are believed to be in the company of their parents, Claire Watson, 32 and 39-year-old Robert O'Neill (39).
The children were reported missing after council officials became concerned for their welfare after failing to contact the family.
Willie Boy is described as being of slim build with fair hair, whilst Tojo is stocky with fair hair.
Since being reported missing to police last week, officers have made extensive inquiries in England - where there are family connections - to trace the family.
So far, there have been no positive sightings of the children, except for the whole family being seen together within the Rivergate Shopping Centre in Irvine at 14:45 on Sunday 3 August.
Police have appealed for anyone with information about the children's whereabouts to contact them urgently.
Police are attempting to trace two children who have been missing from the Irvine area of North Ayrshire.