David Cameron ramps up case for UK action as he suggests support of Syria's president is not needed under international law

David Cameron has for the first time opened a legal path to strike Islamic State (Isis) inside Syria by saying Bashar al-Assad's government is illegitimate.

He suggested the west would not need an invitation from Assad under international law to strike at Isis within Syrian borders.

Speaking at the start of the Nato summit in Wales, Cameron ramped up the case for UK involvement in air strikes in Iraq, saying that Isis represented a direct threat to the UK – and that decisions on strikes would be taken if they were in the national interest.

The prime minister, speaking in a round of broadcast interviews ahead of a meeting with the US president, Barack Obama, also disclosed he wanted to do more to arm Kurds, as well as potentially even train some of their battalions so they can defend their minorities and people.

Cameron is still treading cautiously – aware of the need to bring public and political opinion with him, as well as to ensure a regional coalition is in place determined to defeat Isis with the support of largely western-led air power.

America has launched as many as 140 air strikes but British air power has so far only been involved in humanitarian aid and some reconnaissance. But Cameron's remarks suggest a case for British involvement is being assembled.

In his interviews on Thursday morning, he repeatedly said it was necessary to learn the lessons of the past, adding he was not seeking to impose a solution over the heads of the countries in the regions, but instead to build an international coalition. He added there was "a crying need for the Iraqi government" to be reformed so that it was broadly based and non-sectarian.

Asked if he now supported air strikes he said: "I certainly don't rule anything out and I absolutely do think that Islamic State is a direct threat to the United Kingdom."

He pointed to Isis-inspired plots in Europe, including the murder of innocent people killed in a Jewish museum in Brussels. "We face a direct threat from this organisation and we must work with partners to put a fatal squeeze on the organisation, but it must start with helping those on the ground that are fighting this organisation."

He added: "We are looking at directly supplying arms to the Kurds and indeed helping to train some of their battalions so we make sure they can defend the minorities and their people."

Asked if he thought the west should cooperate with Assad in Syria so as to attack the headquarters of Isis inside Syria's borders, Cameron argued: "President Assad is part of the problem, not part of the solution."

He said: "Assad's brutality gave credence to IS [Islamic State]." In Iraq, he said, the same thing had happened "because there was an Iraqi government that was standing up for the Shias and not the Sunnis and the Kurds that again left a space for this poisonous organisation to fill".

Asked if he needed to make a pragmatic deal with Assad in the face of the greater Isis threat, he said: "In the past just simply saying 'my enemy's enemy is my friend' has led to all sorts of moral quagmires and difficulties. Assad has been part of the creation of Islamic State rather than being part of its answer."

In terms of the legalities of air strikes and the need to be invited by a sovereign country to make such strikes he said "the Iraqi government was legitimate", but in Syria: "President Assad has committed war crimes on his own people and is therefore illegitimate. We would not do anything without moral or legal justification."

He added he was personally supervising the efforts to rescue the British hostage seized by Isis and now under threat of execution, saying his heart went out to their family, "We do everything we can in every circumstance and ask what more can we do," he said.

"There was an attempt at hostage rescue some weeks ago that was sadly not successful, but it is right we should not pay ransoms to terrorists in these circumstances.

"I am convinced that when these ransoms have been paid and tragically they have been paid in some circumstances the money goes directly to kidnapping more people getting arms and weapons and plotting more terrorist outrages including here in the UK."

He hinted there were efforts being made to communicate with the hostage-takers presumably through intermediaries, but said "there was a world of difference between a communication and paying some ransom".

The Labour leader, Ed Miliband, told BBC Radio Scotland's Good Morning Scotland programme that he would look at the case for air strikes on its merits "People across our country have been shocked at the brutality of Isis, not just against British citizens, but against people from right across every community," he said.

"It's a threat which can't be ignored; I think its very, very important that we don't just turn away from it and say 'it's too big a problem'.

"It also means we've got to learn the lessons of the past. I think that means first of all we've got to build an alliance within the region – it's not just about Britain and the United States, its about countries within the region, Saudi Arabia, Turkey, Qatar."

Nick Clegg, the deputy prime minister, sounded a very cautious note on the possibility of Britain joining air strikes on Isis, saying it would be a "total disaster if this comes to be perceived as the west against the rest".

"We want to keep ourselves safe, but we are acting through the humanitarian aid that we have provided, the air strikes that the US are conducting, in support of the legitimate government of Baghdad, the legitimate authorities in the Kurdish region against a terrorist organisation, which is literally creating, carving out a new country across two other nations," he told LBC 97.3.

"Let's absolutely get away from this idea that we can fix this from Washington or we can fix this from the Ministry of Defence.

"This is part of a much wider movement which has to be led first and foremost by the countries and legitimate authorities in the region. The moment this collapses or any one allows this wittingly or otherwise to collapse into a perceived west versus the Islamic world – I think we would then be drawn into a terrible downward spiral of violence, from which it would be very difficult to escape."