David Cameron will signal this week that he hopes to maintain Britain’s membership of the European Union when he challenges a series of claims by campaigners seeking to take the UK out of the EU.

The prime minister will use a visit to the Northern Future Forum in Iceland to dismiss claims by the Leave group that Britain could enjoy a special relationship with the EU on the lines of the accord negotiated by Norway.

Cameron will also reject a proposal supported by Boris Johnson for two EU referendums. The first one would be held, as planned by the prime minister by the end of 2017. But the second would be held at a later date on a new relationship between the UK and the EU after an initial no vote.

“The referendum will be definitive,” said one government source. “If the people of Britain vote to leave then we leave. If we vote to remain then we remain.”

The intervention by No 10 is a rebuff to the London mayor who has expressed interest in the idea of two referendums proposed by Dominic Cummings, one of the main strategists in the Leave campaign who used to serve as a special adviser to Michael Gove.

The Leave campaign has also sought to downplay the risks of a British exit by following the example of Norway, which has access to the EU’s single market as a member of the European Economic Area. But Norway pays hefty fees, and has no say over the rules of the single market. The group also cites the example of Switzerland, which negotiates access to the single market.

During his visit to Reykjavik, the prime minister will repeat his criticism of the so-called Norway option that he made in his speech to Bloomberg in January 2013, in which he set out his plans to hold a referendum. He said in the speech: “There are some who suggest we could turn ourselves into Norway or Switzerland – with access to the single market but outside the EU. But would that really be in our best interests?

“I admire those countries and they are friends of ours – but they are very different from us. Norway sits on the biggest energy reserves in Europe, and has a sovereign wealth fund of over €500bn. And while Norway is part of the single market – and pays for the principle – it has no say at all in setting its rules: it just has to implement its directives.

“The Swiss have to negotiate access to the single market sector by sector. Accepting EU rules – over which they have no say – or else not getting full access to the single market, including in key sectors like financial services.”

The prime minister’s planned remarks show he is preparing to up the tempo after the Polish elections which saw victory for the highly conservative Law and Justice party. Iceland is not a member of the EU, but the prime minister will use the forum to lobby the Baltic states and Scandinavian members of the EU.

The pro-EU campaign will launch a poster campaign highlighting new research that shows that every person in Briton would still pay £115 a year to the EU under the Norwegian option. Will Straw, executive director of Britain Stronger in Europe, said: “While we know that leaving Europe would damage our economy, we cannot be certain of the scale of the damage. Vague assertions from both of your campaigns that Britain could prosper from standing alone from our largest trading partner are insufficient in a debate of this magnitude.”