Willie Boy O'Neill, three, and Tojo O'Neill, two, are believed to be with their parents Claire Watson, 32, and Robert O'Neill, 39.
The children, formerly of Caldon Road in Irvine, North Ayrshire, were reported missing from the area last week.
Police said concerns had been raised for their welfare after attempts by North Ayrshire Council to engage with the family before they went missing "met with negative results".
The family were last seen together at the Rivergate Shopping Centre in Irvine at 2.45pm on Sunday August 3.
Chief Inspector Martyn Hopkins urged the children's parents or anyone with knowledge of their whereabouts to contact Irvine police by calling 101.