Witnesses described seeing tanks and troops in the centre of the capital, Khartoum, early on Thursday.
Sudan has seen demonstrations in recent months over the high cost of living and demands for political reform.
President Omar al-Bashir has been in power since staging a coup in 1989.
Information Minister Ahmed Belal Osman said 13 people had been arrested over a plot which “targeted the stability of the state and some leaders of the state”, AFP news agency reports.
“The most prominent is retired general Salah Gosh,” he said.
Arab Spring
Armoured vehicles and tanks were seen in the early hours of Thursday in a main street in central Khartoum, although news agencies said there was no increase in security later on.
Mr Osman said military and state intelligence officers had also been arrested in connection with the “plot”.
“The authorities decided to stop them before they implemented it,” he added.
Mr Gosh was a close ally of President Bashir, and served as chief of the intelligence service until 2009.
Although Sudan has witnessed protests in recent months calling for the end of Mr Bashir’s 23-year regime, they have since died out after a number of opposition figures were arrested.
Pro-democracy activists had hoped to organise protests similar to those that swept long-serving Arab rulers out of power in North Africa.
Mr Bashir was also unpopular with Sudanese nationalists over his decision to give South Sudan – the source of most of its known oil-reserves – independence last year, correspondents say.
The decision has badly hit Sudan’s economy, with growing anger over the spiralling cost of food and other basic items, they say.
There have also been clashes along the Sudan and South Sudan borders, and the two countries have been locked in a dispute over oil revenues.
In September, Sudan agreed with South Sudan to halt fighting and resume oil exports from the South via Sudan.
However, the two neighbours have failed to implement the terms of the accord and South Sudan has accused Sudan of carrying out air strikes over the border, an accusation denied by Khartoum.
The Sudanese army said it was targeting rebels from the Darfur region well inside its territory and accused the South of supporting the rebels.
Earlier this month, another rebel group – the Sudan People’s Liberation Movement-North – said it had shot down a government military aircraft on the disputed border between Sudan and South Sudan.