Mr Bush and Mr Koizumi get along well

The two leaders spoke during closed-door talks in Tokyo on the first leg of Mr Bush's six-country tour of Asia.

Mr Bush, who has spoken warmly of his friendship with Mr Koizumi, was expected to have raised US concerns about the value of the Japanese yen and its impact on US exports.

Iraq was also high on the agenda, following a pledge from Japan this week for an initial $1.5bn in grant aid for the reconstruction of Iraq during 2004.

BUSH IN ASIA Japan agenda: US complaints about the weak yen hurting US exports Funding and troops for Iraq Consultation on N Korea's nuclear ambitions Rob Watson's reporter's diary

"Really frank. Meaningful. And interesting. Fantastic," Mr Koizumi said of their meeting as he posed for photographs with Mr Bush and his wife Laura.

Mr Bush said of Mr Koizumi: "He's a good friend. A very strong leader. And the relationship between Japan and the United States is very good."

Mr Bush and Mr Koizumi were trying to build on a relationship cemented during the Japanese leader's trip to the US earlier this year.

As well as Japan, President Bush will visit the Philippines, Thailand, Singapore, Indonesia and Australia over the next eight days.

In Thailand he will join other regional leaders at the annual Asia Pacific Economic Cooperation (APEC) forum.

The war on terrorism and funding for Iraq are expected to be high on the agenda.

But, says the BBC's Rob Watson travelling with Mr Bush, the issues that may grab most of the headlines are trade and currency wars.

Many business people and workers in America accuse Japan and China of keeping their currencies low to make it easier to sell products in the US, leading to a loss of jobs and exports.

Market-led

Before leaving for Asia, President Bush promised he would tackle the issue.

While expressing optimism over the US economy, he said: "We need a level playing field when it comes for trade and a level playing field will help us create jobs here in America."

Security will be important throughout the trip

A senior official on board Air Force One told reporters that Mr Bush's position was that the markets should determine the strength of a currency.

"The markets act to set the exchange rate. We have a strong dollar policy," the official said.

However, it will be a difficult issue for the president to raise given his need for co-operation over Iraq and North Korea, says our correspondent.

He also says it is no accident that Mr Bush will be stopping off in the Philippines, Singapore and Australia. They are all supporters of his foreign policy.