The UK should be prepared to use its diplomatic and military prowess to help defeat the "monstrous organisation" of the Islamic State (Isis) in northern Iraq, David Cameron has said.

As Britain steps up its military involvement – by flying warplanes deeper into Iraq in a mission that could last months, according to the defence secretary, Michael Fallon – the prime minister warned of an extremism crisis in the country.

But Cameron, who confirmed that Britain would look favourably on any request for arms from Kurdish forces fighting Isis, told BBC1's Breakfast programme on Monday: "Britain is not going to get involved in another war in Iraq. We are not going to be putting boots on the ground. We are not going to be sending in the British army.

"Yes we should use all the assets we have – our diplomacy, our political relationships, our aid, the military prowess and expertise we have to help others – as part of a strategy to put pressure on Islamic State and make sure this terrorist organisation is properly addressed and it cannot cause mayhem on our own streets."

Cameron later rejected calls to recall parliament to discuss the Iraq crisis. He said he would keep the issue of a parliamentary recall under review but insisted that Britain's response to the humanitarian crisis in Iraq did not merit an emergency Commons debate.

Speaking at the Relationships Alliance, Cameron said: "I don't think it is necessary to recall parliament. I think that we are not contemplating things that would require that. But I am very happy to keep that under review, as I always do whenever there is a parliamentary recess. We have parliamentary recalls from time to time."

The prime minister also defended his decision to resume his summer holiday, and made clear that he never handed the reins of government to a deputy – remarks that were interpreted by Ukip as a snub to Nick Clegg.

He said: "In terms of making sure that the government is properly represented at all levels, I always make sure there are senior ministers on duty in Westminster. But I don't hand over the government to a deputy. Wherever I am in the world, I am always within a few feet of a Blackberry and an ability to manage things should they need to be managed.

"And indeed, as I have done on almost every holiday I have enjoyed over the last few years, to return instantly should that be necessary. For the next few days I won't be terribly far away. So if that is necessary you'll find me at my desk."

Fallon said Britain's involvement in Iraq was fast expanding beyond the initial humanitarian mission to relieve Yazidi refugees besieged on Mount Sinjar.

The defence secretary confirmed that RAF Tornado jets, which were deployed to the region last week to identify the location of Yazidi refugees, were now monitoring Isis positions deep into Iraq.

Cameron also confirmed that Britain had moved beyond providing humanitarian relief. He told the BBC: "We are trying to help with the situation we face. First of all that is a humanitarian crisis in Iraq, with people being displaced and British aid and British military have been able to help deliver aid to people including on Mount Sinjar but also now increasingly to the refugee camps.

"But alongside the humanitarian crisis there is also a political and extremism crisis in Iraq that has a direct effect on us back here in the UK. We do have a fully worked through strategy for helping with allies to deal with this monstrous organisation – the Islamic State."

The US announced its most concerted bombing campaign yet over northern Iraq to drive Isis forces back, helping Kurdish forces to reclaim Mosul dam. American warplanes and drones carried out more than 20 strikes at the weekend near the crucial dam that Isis forces took earlier this month. More than a dozen Isis vehicles were destroyed, the Pentagon said. The bombing is designed to help Kurdish forces regain the initiative in fighting for territory that Isis has seized over the past three months.

"Mosul dam was liberated completely," Ali Awni, an official from Iraq's main Kurdish party, told Agence France-Presse. That statement was confirmed by another party official and a Kurdish security forces officer.

Britain was encouraged by the progress over the weekend. But Downing Street believes that the US, the Kurds and forces of the Iraqi state face a lengthy battle to defeat the Isis jihadis.

Fallon's declaration that British involvement was expanding beyond a humanitarian mission contrasted with the cautious language of Downing Street last week when officials said that the British mission was limited to providing humanitarian relief for Yazidi refugees. Britain announced towards the end of last week that it would be prepared to arm Kurdish forces fighting the extremists.

Fallon said: "We and other countries in Europe are determined to do what we can to help the government of Iraq combat this new and very extreme form of terrorism that [the Islamic State] is promoting."

The defence secretary was speaking at RAF Akrotiri in Cyprus as six Tornado fast jets and a spy plane pushed beyond the Kurdish region in the very north of Iraq – the focus of the humanitarian crisis – into parts of mainland northern Iraq that were taken over by Islamist militants in the past two months.

The aircraft are providing footage and other intelligence on displaced families to assist with the humanitarian effort and also on the movement of jihadis on the ground. It is studied by British analysts before being used to help Kurdish and Iraqi security forces.

British intelligence is shared directly with the US military, which has been conducting air strikes against Islamic State positions for the past 10 days.

"We are helping the Iraqi authorities and the Kurdish forces build up a better picture of where Isis is, where it is likely to strike next," Fallon said in an interview conducted by the defence editor of the Times and pooled on behalf of the major newspapers including the Guardian.

This means that British information could be used by the Iraqi military in planning attacks against jihadis – a development that brings Britain closer to a direct combat role, which could cause alarm among some MPs who are concerned about mission creep.

The Rivet Joint spy plane, a sophisticated, electronic surveillance aircraft, has been conducting missions since last week. Its information is combined with intelligence from daily patrols conducted by pairs of Tornado fighter jets, equipped with surveillance pods.

The RAF is also flying planeloads of ammunition, rifles and machine guns from former Soviet bloc countries in eastern Europe into Irbil in order to strengthen the fighting power of the Iraqi Kurdish security forces who are combating the jihadis.

In addition, the government is looking at what British gear, such as night-vision goggles and body armour, it could send directly to the country to boost the Kurdish peshmerga fighters.

The defence secretary praised a group of about three dozen air force personnel and soldiers from the various different units at the RAF base for their work in distributing aid. Fallon then signalled an expansion of the mission which could last months when he said: "This mission isn't over. The humanitarian needs are there … There may well now be in the next few weeks and months other ways that we may need to help save life, protect people. We are going to need all of you again and the surveillance you are able to give us."

In his interview, the defence secretary confirmed that a small number of regular British soldiers were sent briefly into the semi-autonomous Kurdish north of the country. The troops from 2nd Battalion the Yorkshire Regiment were flown into Irbil, the capital of Iraqi Kurdistan, last week to prepare for a possible deployment of Chinook helicopters on a mission to save Yazidi families.

Deborah Haynes, defence editor of the Times, despatched a pooled contribution to this report from RAF Akrotiri in Cyprus