David Cameron will risk provoking the Tory eurosceptic right on Monday when he joins forces with the arch pro-European Kenneth Clarke to argue that British membership of the EU remains a vital national interest.

In a speech explaining Britain's standing in the world a week before he hosts the G8 summit in Northern Ireland, the prime minister will say that membership of the EU is crucial in guaranteeing Britain a seat at the "top table".

Cameron's staunch defence of Britain's EU membership, a month after Michael Gove and Philip Hammond said they would vote to leave now, will be reinforced by Clarke who will warn that Britain will be "reduced to watching from the sidelines" if it leaves the EU.

The prime minister will indicate his sympathies lie with Clarke and not with his friend Gove when he outlines how Britain can improve its standing in the world.

Cameron will say in a speech in Essex that Britain has a "place at the top table" as a result of its seniority at the UN, the Commonwealth, Nato, the WTO, the G8, the G20, adding "and yes – the EU".

The prime minister will say: "Membership of these organisations is not national vanity – it is in our national interest. The fact is that it is in international institutions that many of the rules of the game are set on trade, tax and regulation. When a country like ours is affected profoundly by those rules, I want us to have a say on them."

But he will say that Britain should be prepared to stand its ground, as he did by negotiating a cut in the EU budget. He will say that enthusiastic membership of the EU "doesn't mean supinely going with the flow of multilateral opinion – the lowest common denominator approach to democracy, as we've seen in the past".

Cameron's argument will be echoed by Clarke, his trade envoy, who will release a paper which shows that the proposed EU-US free trade agreement would boost UK car exports by 25%. He will say that the agreement would boost total UK exports by 1.3% or £19bn.

Clarke will have a stern warning for Tory eurosceptics who would like to leave the EU. "It [the free trade agreement] is … the sort of world-defining reform that we, in the UK, could only ever lead from within the EU. The prime minister has been banging the drum from inside the EU to ensure that this agreement is the best it can be for Britain. Outside the EU we would be reduced to watching from the sidelines."

Cameron will say that his approach to the EU – effectively acting as a critical friend – symbolises his middle ground approach to global relations in which the UK has an outward-looking perspective but is prepared to stand its ground. He will say: "There are wrongheaded approaches we must avoid. The rejection of the modern world and the un-questioning embrace of globalisation – they amount to different kinds of national timidity. Either too wary to engage with the world or too afraid to stand up for our national interests.

"My argument – and the argument of this government – is that to succeed, it's no use just hiding away from the world; we've got to roll our sleeves up and compete in it. And it's no use just giving in to the world – we've got to be unashamedly bold and hard-headed about pursuing our national interests."

But the prime minister will warn that Britain's prospects are being hampered by "national weaknesses" such as a "bloated" welfare system. He will say: "We have identified, very clearly, our key areas of national weakness compared to the rest of the world. One: our debt-fuelled, unbalanced economy. Two: our bloated welfare system. Three: our under-performing education system.

"These are the priorities that define and drive our domestic agenda. A stronger economy. Welfare that works. A world-class education system. And we are pursuing them with ruthless ambition for everyone in this country."