The timing is exquisite. While Conservative Cabinet ministers at home talk publicly about leaving the European Union, David Cameron in the United States will talk about how much it matters.

The prime minister is coming to the White House to make the case for a new EU-US trade deal, a free trade area that he says could mean £10bn a year to the British economy. He calls it a once-in-a-generation prize that could help build a more dynamic world economy.

The deal would cut tariffs and trade barriers and harmonise regulation so that a product approved in the UK could automatically be sold in the US. Mr Cameron says he hopes to give the deal a boost by starting formal talks with President Obama at next month's G8 summit in Northern Ireland. But those talks could take some years, years that presume Britain's continued membership of the EU, something that the US has said it wants to continue, and which many Conservatives do not.

From his body language and tone of voice, the prime minister is clearly irritated by the growing debate about Europe. While he visits Washington, Boston and New York, Tory MPs are clamouring for a debate on the UK's future relationship with the EU. He knows he will have to shout loudly to get his voice heard on the other issues he wishes to raise here on his three day trip. But raise them he will:

1. Syria. The PM hopes to put what he calls "flesh on the bones" of the plan for some kind of international peace conference. He says his talks with President Putin of Russia last week were "a positive and good conversation" and there was a "recognition that it would be in all our interests to have a safe and secure Syria". He will discuss the next steps with President Obama. He says: "We have a long way to go. But I am looking to turn this proposal into a peace conference and that will make a real difference."

2. The G8 summit. The UK is currently holding the G8 presidency and the trade bloc's leaders will gather in Northern Ireland next month. The key issue is Mr Cameron's attempt to persuade countries to be more transparent about their tax systems and agree to crack down on tax evasion. The aim is to encourage developing nations to open up their markets to the world by giving them confidence that they will get decent tax revenues as their economies grow. Some NGOs say Mr Cameron will only achieve this if he tackles tax transparency in British dependent territories. But he says: "I am confident we are going to make progress on this," and is looking to get what he calls "America's full support for this agenda".

3. Boston bombings. Mr Cameron will visit FBI headquarters in Washington to discuss what Britain can learn from the way the US authorities respond to bombings and other major incidents. Alongside the new head of MI5, Andrew Parker, the PM will see the FBI's strategic operations centre that runs around the clock, and has 400 staff and more than a thousand phone lines. And we have Cabinet Office Briefing Room A, otherwise known as Cobra. "We need to learn the lessons of what happened in Boston and compare notes on the issue," he said.

4. Development goals. On his trip, Mr Cameron will visit the UN to discuss his latest plans to update the millennium development goals. He says he wants to "nail down some simple commitments that everyone can get behind".

That, at least, is Mr Cameron's agenda. But it will be Europe that will follow him around like a bad smell on this trip. And he won't be able to hold his nose forever.