The Daily Sport and Sunday Sport have suspended publication and will go into administration, their owner has said.
Sport Media Group, which was saved from going out of business by former owner David Sullivan in 2009, said it had ceased trading with immediate effect.
The announcement came after the group warned it had experienced "an insufficient recovery" since December.
The Daily Sport was launched by Mr Sullivan in 1991, following on from the Sunday Sport, first published in 1986.
If no buyer is found, the Sport will be the first UK national newspaper to close since the demise of News International's Today in November 1995.
'Bomber on moon'
The Daily Sport was relaunched in April 2008 under the editorship of Barry McIlheney and James Brown - the latter being the founder of Loaded magazine - with the stated aim of going from "sleazy" to "sexy".
"If it is not about sport, if it is not about girls and does not make you laugh, then don't bother," Mr McIlheney said at the time, explaining his editorial policy.
As part of the relaunch, former Liberal Democrat MP Lembit Opik began a weekly political column for the paper.
Further back, both papers were edited by Tony Livesey, who now presents a programme on BBC Radio 5 live.
Under his tutelage, the Sunday edition in particular became known for its untrue but outlandish stories - such as "World War Two Bomber Found on the Moon".
Sport Media Group (SMG) said on its website that it had ceased trading because of its "inability to meet certain creditors as they fall due" and was in the process of appointing administrators.
The group suspended trading in its shares on Friday morning "pending clarification of its financial position".
An "insufficient recovery" in trading had occurred since the adverse weather in December 2010, it said.
The Daily Sport's circulation peaked in 2005 at 189,473, while the Sunday edition reached a high in the same year of 167,473.
But SMG withdrew its titles from the official newspaper industry monthly circulation audit in 2009 after sales plummeted.
The National Union of Journalists' assistant organiser for the North of England, Lawrence Shaw, said the company had called an all-staff meeting for Monday lunchtime.
In its most recent annual report - in 2009 - SMG said it had 131 employees.