Last year, Ansar Bayt al-Maqdis claimed responsibility for a failed assassination attempt on the Interior Minister.
The army-backed government, along with state-run and most private media, accuse Mursi's Brotherhood of carrying out terrorist attacks.
Authorities have tightened security around Cairo in anticipation of trouble, as rival political factions plan to gather to mark the anniversary of Mubarak's downfall.
(Read more: Death toll mounts in Egypt as 'civil war looms')
Police and army have deployed heavily, and helicopters flew over a ghostly quiet downtown Cairo on Saturday morning.
Mohamed Saleh, A 63-year-old taxi driver said he planned to commemorate 2011 revolt by going to Tahrir Square, the symbolic heart of the uprising, but the square was tightly sealed off by armoured vehicles on Saturday morning.
"This was a complete popular uprising by the people," Saleh said, recalling the historic 18 days of protests that led to Mubarak's ouster. "It was protected by the army," he added.
Clashes in the capital and several other cities on Friday between Mursi supporters and security forces killed 14 people, in another sign of high tensions in the biggest Arab nation.