Activists for and against ousted Egyptian president Mohamed Morsi have called rival rallies for the first Friday of Ramadan, as tensions soar over the army's overthrow of the Islamist leader.

The Muslim Brotherhood, the influential group from which Morsi emerged, has vowed to keep protesting until he is reinstated.

The group has called separate rallies across Cairo, but the group's ability to mobilise remains in question with much of its leadership detained, on the run or keeping a low profile.

The anti-Morsi camp has also called for rallies, including a mass iftar -- the breaking of the Muslim fast -- in Tahrir Square.

The military's overthrow of Morsi last week after millions took to the streets demanding his resignation has sparked deadly clashes and deepened divisions in the Arab world's most populous country.

Egypt's new premier said on Thursday he did not rule out Muslim Brotherhood members in his cabinet, even as the group vowed to keep defying the coup.

Hazem al-Beblawi, who was appointed on Tuesday, told AFP in a telephone interview: "I don't look at political association...

"If someone is named from (the Brotherhood's) Freedom and Justice Party, if he is qualified for the post" he may be considered.

"I'm taking two criteria for the next government. Efficiency and credibility," he added.

Beblawi said he wanted to decide on the best candidates before asking them to join the government.

The Brotherhood has already rejected a Beblawi offer to join the new government. They said the mass rally called for Friday was against what it called "a bloody military coup".

The United States meanwhile said it was pressing ahead with plans to deliver four F-16 fighters to Egypt, a US official said.

There was no decision to halt the scheduled transfer of the warplanes or to cut off other security assistance to Egypt, the official said on condition of anonymity, even though Washington has announced a review of all aid to Cairo.

President Barack Obama's administration has said it is examining whether the military takeover constitutes a coup, which under US law would force Washington to freeze any aid to Egypt.

A State Department spokeswoman said Thursday that previously planned assistance would continue to flow to Egypt.

And the Pentagon said it remained dedicated to maintaining longstanding military ties with Egypt and that the United States wants to see a prompt return to civilian, democratic rule.

In Cairo, the bloody run-up to Ramadan has marred the onset of the holy month.

In the worst incident, clashes around an army building on Monday left 53 people dead, mostly Morsi supporters.

In the restive Sinai peninsula, a Coptic Christian man was found decapitated on Thursday five days after gunmen kidnapped him, security officials and witnesses told AFP.

Police were hunting the Brotherhood's Supreme Guide, Mohammed Badie, after a warrant was issued for his arrest on Wednesday.

Badie and other senior Brotherhood leaders are wanted on suspicion of inciting the clashes.

Last week Badie gave a fiery speech in which he vowed that Brotherhood activists would throng the streets in their millions until Morsi's presidency was restored.

Morsi himself is currently being held in a "safe place, for his safety", foreign ministry spokesman Badr Abdelatty told reporters on Wednesday, adding: "He is not charged with anything up till now."

Military and judicial sources have said the ousted leader may eventually face charges.

UN leader Ban Ki-moon raised concerns about the detention of Morsi and other Muslim Brotherhood leaders with the country's foreign minister, a spokesman said.

Ban reaffirmed that there must be no "retribution" against any party, UN spokesman Martin Nesirky told reporters after the telephone talks between Ban and Foreign Minister Mohammed Kamal Amr.

While the Brotherhood accuses the army of "massacring" its supporters in Monday's incident, the army says soldiers were attacked by "terrorists" and armed protesters.

The public prosecutor pressed charges on Wednesday against 200 of 650 people detained during the violence.

Adly Mansour, the military-appointed caretaker president, has set a timetable for elections by early next year.

But opponents and supporters of Morsi alike have criticised the interim charter he issued on Monday to replace the Islamist-drafted constitution and steer a transition that the army itself has acknowledged will be "difficult".

Many fear a repetition of the mistakes of the last military-led transition, between Hosni Mubarak's ouster in 2011 and Morsi's election in June 2012.

In Sinai early Friday, militants killed an Egyptian police officer in an attack on a checkpoint with rocket-propelled grenades, security and medical officials said. A police conscript was wounded in the attack.