UPDATE 3-Egypt Islamists protest, politics hit a snag
* Islamists protest peacefully in Cairo
* Confusion over prime minister adds to Egypt's crisis
* No further word on appointment of interim government
* Political progress key after wave of deadly protests
(Releads with protests, adds quotes, colour)
CAIRO, July 7 (Reuters) - Thousands of supporters of Egypt's ousted President Mohamed Mursi protested outside his place of detention in Cairo on Sunday while a military-driven plan to resolve the political crisis remained mired in mistrust and confusion.
Demonstrators at the Republican Guard barracks, where three people were killed on Friday, shouted "Mursi, Mursi, God is greatest!" and "Peaceful, peaceful!" as soldiers and policemen looked on from behind barbed wire.
"We will not leave until Mursi returns. Otherwise we'll die as martyrs," said 55-year-old Hanim Ahmad Ali Al-Sawi, wearing a veil over her face in the searing midday sun. "This was a coup against democracy."
Mursi was toppled on Wednesday in a takeover the military denied was a coup. The army said it stepped in to enforce the will of millions of Egyptians who rallied on June 30 demanding his resignation.
But while Mursi's ouster was met with scenes of jubilation, it angered Islamists who held protests on Friday in which more than 30 people were killed and 1,400 wounded.
The violence across the Arab world's most populous state saw rival factions fighting street battles in central Cairo and many others cities and towns, and underlined the pressing need for a swift and inclusive political solution.
Egypt's allies in the West, including main aid donors the United States and the European Union, and in Israel, with which Egypt has had a U.S.-backed peace treaty since 1979, have looked on with increasing alarm.
The transitional authorities had been set to appoint liberal politician Mohamed ElBaradei, a favourite of young anti-Mursi protest leaders, before his candidacy was thrown into doubt when a hardline Islamist party objected.
The abrupt U-turn due to opposition from the Nour Party, Egypt's second Islamist force after the Brotherhood, highlighted the challenge the military faces in finding consensus among liberals and conservatives on who should run the country and what direction they should lead it in.
"We extend our hand to everyone," a presidential spokesman told reporters late on Saturday. "The Muslim Brotherhood has plenty of opportunities to run for all elections including the coming presidential elections or the ones to follow."
Minutes after he spoke, state media reported that the public prosecutor had ordered four top Brotherhood leaders arrested this week to be detained for a further 15 days on accusations of inciting violence against protesters.
The Brotherhood has said it wants nothing to do with the military's plans for a new interim government. It wants Mursi reinstated and has pledged to keep protesting until he is.
The military has shown no sign of moving to dislodge the Islamists and may be hoping that sweltering summer heat and the onset of the Ramadan Muslim fasting month from Tuesday will gradually wear them down.
The Nour Party, the Brotherhood's rival for the Islamist vote, had agreed to the army-backed transition plan leading to new elections. Its withdrawal from the process would strip that plan of Islamist legitimacy.
For many Islamists, the overthrow of Egypt's first freely elected president was a bitter reversal that raised fears of a return to the suppression they endured for decades under autocratic rulers like Hosni Mubarak, himself toppled in the 2011 Arab Spring uprisings.
The Brotherhood called for more protests on Sunday, although by mid-afternoon they had not matched the numbers who marched two days earlier. Tahrir Square, cradle of the huge anti-Mursi movement, saw only small crowds over the weekend.
On Saturday, President Barack Obama condemned the violence and said the United States was not working with any particular party or group in Egypt.
Washington has not condemned the military takeover or called it a coup, prompting suspicion within the Brotherhood that it tacitly supports the overthrow.
Obama has ordered a review to determine whether annual U.S. assistance of $1.5 billion, most which goes to the Egyptian military, should be cut off as required by law if a country's military ousts a democratically elected leader.
U.S. Defense Secretary Chuck Hagel held three conversations with the head of Egypt's armed forces on Friday and Saturday, emphasising the need for "a peaceful civilian transition in Egypt", the Pentagon said on Saturday.
A U.S. official, speaking on condition of anonymity, said the conversations between Hagel and military leader General Abdel Fattah al-Sisi lasted more than two hours.
Egypt can ill afford to lose foreign aid. The country appears headed for a looming funding crunch unless it can quickly access money from overseas. The local currency has lost 11 percent of its value since late last year.
The governor of Egypt's central bank, Hisham Ramez, flew to Abu Dhabi on Sunday, officials at Cairo airport said, following Egyptian media reports Cairo was seeking financial aid from Gulf states after Mursi was toppled.
Only gas-rich Qatar provided substantial funds to Mursi's government, totalling $7 billion in loans and grants. Turkey and Libya provided smaller loans and deposits.
But Egypt's foreign reserves fell $1.12 billion in June to $14.92 billion, representing less than three months of imports.
Only about half are in the form of cash or in securities that can easily be spent, and the IMF considers three months to be the minimum safe cushion for reserves.
(Additional reporting by Mike Collett-White, Maggie Fick, Sarah McFarlane, Shadia Nasralla, Tom Perry, Paul Taylor, Patrick Werr in Cairo and Steve Holland in Washington; Writing by Mike Collett-White; Editing by Paul Taylor)