Goldsmith later told the cabinet in March 2003 that the war was legal without a UN resolution, and the US-led invasion went ahead three days later.
The letters were released as Hoon gave evidence to a British public inquiry into the Iraq war.
He is the most senior politician to appear before the hearings so far.
Hoon told the hearing in London on Tuesday that Goldsmith had "categorically" concluded that "there was a legal justification for military action" in his final conclusion in 2003.
He also said the government wanted a diplomatic solution to the crisis and did not think Tony Blair, the UK prime minister at the time, gave a promise to George Bush, the US president, to support war come what may.
"I think that right up until the vote in the House of Commons, our attitude towards
the use of force was always conditional," Hoon said.
He identified mid-2002 as the period when it become clear that Washington "meant business" over Iraq, because the country was so deeply traumatised over the September 11, 2001 attacks and it perceived Saddam Hussein, the Iraqi leader at the time, as another threat.
"It was getting pretty real by then. I think there was a real sense of the Americans thinking through in a very practical way the consequences of the 'axis of evil' speech [by Bush] and focusing on Iraq.
"So we had no doubt at that stage in the summer that they meant business."
Bush first branded Iraq, Iran and North Korea as an "axis of evil" in his State of the Union Address on January 29, 2002.
The Iraq war inquiry is Britain's third and widest-ranging inquiry into the conflict, which triggered huge anti-war huge protests in the UK at the time and in which 179 British soldiers died fighting.
Nazanine Moshiri, Al Jazeera's correspondent in London, said the inquiry will be looking to Blair, who is due to testify on January 29, for key evidence at the hearing.
"The Iraq inquiry says it is 'struggling to find' key intelligence evidence supporting Tony Blair's case for war," she said.
"It will be asking him to fill in those gaps when he appears here next week."
Two previous reports on aspects of the war have cleared Blair's government of wrongdoing over the conflict.