U.K. Prime Minister David Cameron will seek during his first official visit to the U.S. to play the role of a pragmatist more eager to drum up trade than indulge in the tributes to the nations' "special relationship" that tend to dominate such trips.

After landing Tuesday, Mr. Cameron will meet with President Barack Obama, Vice President Joe Biden and other senior U.S. officials in Washington, before flying to New York, to meet with United Nations Secretary-General Ban Ki-moon and New York Mayor Michael Bloomberg. Most of the meetings will be devoted to building relationships, with an emphasis on issues such as the war in Afghanistan and trade between the two countries.

Throughout the trip, Mr. Cameron will highlight the latter issue. Britain's new coalition government, which he leads, has placed a major focus on pushing trade as a key part of foreign relations.

In a speech earlier this month, Mr. Cameron told the nation's diplomats: "If you want to keep Britain's great ambassadorial residences—then I want you to show me that every day you are using them relentlessly to open new trade links and to generate new business for Britain."

The U.S. is the U.K.'s biggest single national trading partner, responsible for 18% of British exports. Bilateral trade valued at $170 billion flows between the two countries and the U.K. and U.S. are each other's biggest foreign investor.

Mr. Cameron points out that each day a million people in America go to work for British companies, and vice versa.

A senior Obama administration official said Monday that he doesn't expect the Oval Office conversation to touch much on trade. Rather, he said, expected topics include Iran, the Mideast peace process, the global economy and the war in Afghanistan.

More importantly, the official said, the meeting will be a chance for the two leaders to build their personal relationship. "We want to continue to cement that personal relationship that already was developing," the official said.

"This is kind of a critical test in any respects

," said Fiona Hill, director of the Center for U.S. and Europe at the Brookings Institution, a Washington think tank. "Is there still any oomph there in the British-U.S. relationship? … Can they work out a pragmatic, businesslike working relationship?"

Two current hot-button issues—troubled oil giant BP PLC and the Scottish government's decision last year to free the convicted Lockerbie bomber—are likely to come up over the two days, though perhaps not directly with Mr. Obama. A senior Obama administration official said he doesn't expect BP's troubles to come up in conversation between Messrs. Cameron and Obama. The two leaders are likely to talk about the Scottish government's decision to free Lockerbie bomber Abdel Baset al-Megrahi from prison last year, White House Press Secretary Robert Gibbs said. Both sides have said they believe it was the wrong decision.

The meeting comes as both countries have acknowledged with increasing openness that the world is changing and they are no longer tied to each other in the same way they once were. Just as Mr. Obama is thought to see the importance of the U.K. and Europe fading in a world of emerging powers in Asia, Mr. Cameron often talks of the U.K.'s need to reflect the changing world order by spreading its affections beyond Washington.

Mr. Cameron also wants to differentiate his dealings with Washington from what he sees as the subservient relationship cultivated by the previous U.K. administrations of Tony Blair and Gordon Brown.

"We're living in a new world where the balance of power in different regions is shifting, and the U.S. is strengthening its ties with rising powers," he wrote in an editorial slated to appear in Tuesday's Wall Street Journal. He adds: "Britain is doing the same thing."

Mr. Cameron describes himself as "hard-headed and realistic" about relations between the U.K. and U.S., saying he knows his country is "the junior partner."

For Mr. Cameron, the U.S. relationship isn't about "historical ties or blind loyalty," but is a partnership of choice "that serves our national interests."

Mr. Cameron displays little of the sentimental attachment to the U.S. of previous senior British politicians, many of whom were partly educated in the U.S., vacation there and cultivated deep relationships with U.S. politicians. Mr. Cameron, who has emphasized domestic policy over foreign affairs during most of his political career, is less known among the Republican Party, which is often twinned ideologically with his Conservatives.

In the coalition agreement the Conservatives struck with the Liberal Democrats after failing to win a majority in May, establishing a "new 'special relationship' " with India is listed alongside maintaining a "strong, close and frank relationship" with the U.S.

Both sides expect to talk about the two main arenas in which the two countries work closely together: Afghanistan and sanctions against Iran. On Iran, Mr. Obama will encourage the European Union to adopt tough sanctions of its own.