* Senators call for Egypt dialogue, release of prisoners
* McCain, Graham meet army chief, vice president, Brotherhood
* US, EU, Arab mediation efforts ease tension, no breakthrough (Recasts with McCain/Graham comments at news conference)
By Maggie Fick and Yasmine Saleh
CAIRO, Aug 6 (Reuters) - Two senior U.S. senators delivered a strong message on Tuesday to the Egyptian military, saying it should release political prisoners, start a national dialogue and return the Arab nation swiftly to democratic rule.
Republicans Lindsey Graham and John McCain were sent to Cairo by U.S. President Barack Obama to help resolve the crisis brought on by the army's overthrow of elected Islamist President Mohamed Mursi.
They urged Mursi's Muslim Brotherhood, many of whose leaders have been jailed including the deposed president himself, to avoid resorting to violence and to join a dialogue on a political way forward.
The two men's mission reflected Washington's anxiety at events in Egypt, a bulwark of its Middle East policy and the first Arab state to make peace with Israel.
But their comments after meeting army chief General Abdel Fattah al-Sisi, interim Vice President Mohamed ElBaradei and interim Prime Minister Hazem el-Beblawi fell well short of an endorsement of their actions.
"The people who are in charge were not elected. The people who were elected are in jail. The status quo is not acceptable," Graham told a news conference.
They also described Mursi's overthrow as a coup - a definition that is hotly disputed by the rival Egyptian sides and among U.S. officials, and could trigger a cut-off off in the $1.3 billion U.S. military aid Egypt receives each year.
However, "cutting off aid would be the wrong signal at the wrong time," McCain said.
Addressing both sides, he said: "If you think you can restore legitimacy by violence, you are wrong. Violence will only marginalize you. If you think you can only negotiate with people in jail, that's a huge mistake."
A spokesman for the interim government, Sherief Shawki, gave a cool response to the senators' words.
He rejected their characterisation of Mursi's overthrow as a coup and said the new authorities, installed by the army, had spelled out a plan for a political transition and new elections.
"There is a roadmap which means that what happened was not a coup and that it was Egyptian people who decided on the roadmap put (forward) by the military and which represents the Egyptian people. We don't want foreign intervention to be imposed on us."
The government would stick by that plan, he said. He also rejected the call to release jailed Brotherhood members, saying they would be dealt with by the courts.
Egypt has been dangerously divided since the overthrow of Mursi on July 3 following huge demonstrations against his rule.
He became Egypt's first freely elected president in June 2012, 16 months after the overthrow of U.S.-backed strongman Hosni Mubarak, who had ruled for nearly 30 years.
Mursi is now being detained at an undisclosed location and thousands of his supporters remain camped out at two protest sites in Cairo which the government has pledged to break up.
Almost 300 people have been killed in political violence since the overthrow, including 80 shot dead by security forces in a single incident on July 27.
A diplomatic push led by envoys from the United States, the European Union, Qatar and the United Arab Emirates has so far helped to hold off further bloodshed between Mursi's backers and the security forces but not achieved a breakthrough.
McCain said the senators also met members of Mursi's Freedom and Justice Party, the political arm of the Muslim Brotherhood.
"All parties should be part of a national dialogue and reconciliation is the only way to bring out peace in this country, but also in order to take part in that national dialogue those parties should renounce violence," he said.
The crisis has put U.S. policy in a quandary. Mubarak was a close ally who kept Islamist militants under heel and maintained peace with Israel.
Washington was slow to support the popular uprising that ousted him and cautiously welcomed Mursi's election.
But fears that Mursi was trying to establish an Islamist autocracy, coupled with a failure to ease economic hardships afflicting most of Egypt's 84 million people, led to mass street demonstrations, triggering the army move.
On Monday, U.S. Deputy Secretary of State William Burns and European Union envoy Bernardino Leon met jailed Brotherhood deputy leader Khairat El-Shater in the prison where he is being held.
They tried to persuade him to recognise that there was no realistic prospect of Mursi being reinstated and to accept a political compromise. A Brotherhood spokesman said Shater had insisted they should be talking to Mursi and the only solution was the "reversal of the coup".
Government political adviser Mostafa Hegazy said on Tuesday the authorities would have to deal with the protesters at the Brotherhood camps at Rabaa al-Adawiya and al-Nahda Square.
"The crowds exist not to find a solution or to enter political life but to disrupt everyday life and endanger the future of the nation," he told the MENA state news agency.
Security forces have promised the protesters safe exit if they quit the camps but have warned their patience is limited.
It is thought unlikely that they would take action before Sunday, the end of Eid al-Fitr, the holiday that marks the close of the Muslim fasting month of Ramadan. (Writing by Angus MacSwan; Editing by Mike Collett-White)