The backlog of 153,000 notifications could potentially have meant that thousands were wrongly allowed to remain in the country, the Chief Inspector of Borders and Immigration said.
The tip-offs were all reviewed by May this year, but Mr Vine said it must be an "ongoing priority" to make sure checks are carried out.
His team examined work at three UKBA offices in Sheffield, Beijing and Delhi.
More top news
The tin needed for a Mary Berry dish and the rules of mouse racing were among questions council call centres dealt with in the past year.
Scientists believe they have invented an ice cream that will melt slower in the sun.
There were shocks, tears and skimpy outfits at the star-studded MTV Video Music Awards in LA.