By last weekend, almost a quarter of households had still to send in their data.
The Daily Telegraph has reported that the Scottish government is planning to keep the survey open for an extra four weeks.
Scotland’s constitution secretary, Angus Robertson, is due to make a statement at Holyrood this afternoon.
BBC Scotland political editor, Glenn Campbell said an extension was “highly likely” because the Scottish government needs better data to plan future services and could not realistically fine hundreds of thousands of people for not taking part.
The census was held over from 2011 because of the pandemic, but the survey went ahead in England and Wales with high levels of completion.
There are concerns that decoupling the survey from other parts of the UK had reduced awareness.
It is also the first time paper forms were not sent to every household in Scotland.
People were asked to complete the form online, although paper copies could be requested.
The Scottish government has increased publicity in recent days which is believed to have improved participation rates ahead of the 1 May deadline.