The UK and French foreign ministers have said a UN communique drawn up in Geneva on Saturday night to address the escalating conflict in Syria will mean President Bashar al-Assad is "finished" and will have to step down.

The communique, which agreed terms for a transitional authority to oversee the end of violence in the country, was hammered out with the inclusion of Russia and China and called for "clear and irreversible steps" after a fixed time frame.

In a sign of rising hostility, Turkey's military has said it scrambled fighter jets to its border on Saturday after Syrian helicopters flew close to the frontier.

Tensions between the two countries have continued to rise since Turkey said two of its fighter jets were downed by Syria and a rescue plane was shot at during the subsequent search for the pilots.

A military statement on Sunday said F-16 jets were scrambled and sent to the border after the helicopters flew in the area on at least three occasions.

The military said the helicopters flew as close as four miles (6.5km) from Turkey.

Saturday's agreement stated that members of the present Syrian government could be included in the new body and it was initially unclear whether Assad could be part of that transitional government.

However, speaking on Sunday morning, the foreign secretary, William Hague, said Assad would be excluded from any unity government under the terms of the agreement.

The Foreign Office said under the terms of the communique, members of any future Syrian unity government would be agreed only by "mutual consent" from opposition and current regime members. A spokesperson said there was no chance of opposition members agreeing to Assad or his closest allies joining a unity government and that the insertion of the phrase had been an important win for pro-opposition government.

Speaking on the BBC's Andrew Marr show after his return from Geneva, Hague said the international community was some way from resolving the crisis, but added: "We are putting great energy into it."

He added: "Is it deeply frustrating that hundreds of people are dying every week while we talk? Of course it is. I spent 10 hours yesterday talking to the foreign ministers of Russia and China about what we can do.

"We made one step forward that's worth having, which is that we agreed with Russia and China what a transitional government should look like. And that there should be a transitional unity government in Syria, and that should be made up of people from the present government and opposition groups on the basis of mutual consent, which would of course exclude President Assad from that."

The French foreign minister, Laurent Fabius, speaking on Sunday backed Hague's approach. "Even if they say the opposite, the fact that the text says specifically that there will be a transitional government with all powers means it won't be Bashar al-Assad … because it will be people that are agreed to by mutual consent. The opposition will never agree to him, so it signals implicitly that Assad must go and that he is finished," he told TF1 TV station.

The agreement has met with negative reaction from opposition groups fighting the Assad regime, who described it as ambiguous and a waste of time.

"Every day I ask myself, do they not see how the Syrian people are being slaughtered?" a veteran Syrian opposition figure, Haitham Maleh, asked. "It is a catastrophe, the country has been destroyed, and they want us then to sit with the killer?"

Maleh said the Geneva agreement would have "no value on the ground".

The UN special envoy, Kofi Annan, said he hoped to see concrete results towards a settlement within a year and insisted "it is for the people of Syria to come to a political agreement".

Addressing the meeting earlier, the US secretary of state, Hillary Clinton, said Assad and his allies had no place in the transitional process.

The Geneva meeting ended amid reports of fresh violence in Syria, including claims that government forces had overrun Douma city, which has been under siege for several weeks.

Clinton said the conditions set out in the statement offered the best chance of a transition to a democratic post-Assad period, including free elections. She added that the US would meet next week with Syria's divided opposition to try to forge unity between them in line with the Kofi Annan principles.

Russia had previously refused to back a provision that would call for Assad to step down to pave the way for a unity government. The final UN communique, which followed a day of difficult negotiations, appeared to preserve the Moscow "red line" – that the Syrian people should have ownership of the transition process and it should not be imposed from outside.

Opposition from Russia, with the backing of China, has been seen by western powers as one of the main obstacles in the efforts to end the conflict, which has claimed more than 15,000 lives. According to western diplomatic sources, the sticking point during talks has been a paragraph in the text interpreted by the US and the UK as suggesting that Assad would need to step down before the transition.

The emergency meeting was attended by the foreign ministers of the UN security council's permanent five members, as well as representatives of the EU and Arab states, including Syria's neighbour Iraq.

Addressing delegates, Annan warned that the war in Syria risked spilling over into a wider regional conflict of "grave severity". However, at times during the day that seemed a moot point as US officials accused Russia of "stonewalling" and raised the prospect that the talks might fail.

Admonishing the foreign powers present, Annan said the crisis should never have reached this point. "Either unite to secure your common interests or divide and surely fail in your own individual way," he said. "Without your unity, your common resolve and your action now … nobody can win and everyone will lose in some way."