Tax return warnings sent by HMRC
Thousands of taxpayers will receive letters this week urging them to deal with any historical tax returns they have failed to submit.
HM Revenue and Customs' Tax Return Initiative is aimed primarily at higher rate taxpayers who have gaps in their self-assessment record.
Fines will be reduced if taxpayers come forward with details voluntarily in the next month.
Criminal inquiries could come after the deadline, the UK tax authority warned.
"This campaign offers a quick and straightforward way to bring your tax affairs up to date, but time is running out," said Marian Wilson, head of campaigns at HM Revenue and Customs (HMRC).
The campaign primarily targets those who have failed to return a tax return for 2009-10 or earlier, and who pay at the 40% tax rate. The campaign is open to those who pay basic rate tax.
HMRC has written to 7,100 individuals who the authority considers to be most likely to have missed a submission "despite penalties, reminders and statements", urging them to come forward.
Anyone who does so is likely to have to pay the tax and any interest owed, as well as a fine set at the equivalent of 10% or 20% of the tax owed.
In some circumstances, HMRC said it would allow people to spread these payments.
Following 2 October, the tax authority said it would use its legal powers to pursue those who have failed to submit self-assessment forms and have tax payments outstanding.
The fine could then increase to 100% of the tax due.