Circulation of the children's comic hit 2 million copies a week between the 1950s and 1980s, as characters such as pie-eating cowboy Desperate Dan and Korky the Cat became household names.
But the gradual decline of demand for weekly comics from children and the rise of other pursuits, such as video games and the internet, led to an ongoing fall in sales over more recent years.
In the second half of 2011, circulation of The Dandy dropped below 8,000, and insides have now told The Guardian that the comic is likely to close down shortly.
Despite outlasting other children's comics such as Beezer and Topper, various attempts to rejuvenate The Dandy for the modern age have not proved successful.
Previous new strategies tried to include the switch to a fortnightly called Dandy Xtreme, and celebrity issues in 2010 featuring stars such as Harry Hill, Cheryl Cole, Simon Cowell, Jamie Oliver and Jeremy Clarkson.
DC Thompson, publisher of The Dandy, confirmed to the paper that closure of the comic was being discussed as part of an overall review of the magazine business.
A spokeswoman for the firm said that characters such as Desperate Dan would continue to exist online if the print title was axed. Another possible solution could be moving them to the Beano, which is also published by DC Thompson.
"We are carrying out a review of our magazines business to meet the challenges of the rapidly changing publishing industry," the spokeswoman said.
"There are many challenges within the industry at present, but we're excited that the digital revolution has also given us an opportunity to innovate and develop.
"We're confident that future generations will continue to enjoy our much-loved products and characters."
The Cartoon Museum in London is currently preparing an exhibition marking the 75th anniversary of the first edition of The Dandy, which went on sale in 1967, priced at 2p with a free whistle.