Mr Tappin is being held on remand in a jail in New Mexico after being denied bail following his extradition last month. His family and friends, who include the UKIP leader Nigel Farage, have raised a 4,000-signature petition demanding that Mr Cameron debates the US-UK extradition treaty with Mr Obama.
But well-placed sources have told the Telegraph that while Mr Tappin's case "may" come up in discussions during Mr Cameron's two-day trip to the US, it is not on the main agenda.
Mr Cameron's stance over Mr Tappin contrasts sharply with his views on Gary McKinnon, the Asperger's Syndrome sufferer who is fighting extradition for allegedly hacking US military computers, where the Government believes there is a genuine issue.
Mr Tappin, who is president of the Kent Golf Society, came to the attention of US authorities in 2007 after the US Bureau of Immigration and Customs Enforcement (ICE) set up a fake company, Mercury Global, in El Paso, Texas.
According to court papers relating to an unsuccessful appeal by one of Mr Tappin's co-accused, Robert Caldwell, Mr Tappin contacted Mercury in Dec 2005 through an intermediary in the US, "seeking the unlicensed export" of a camera and Hawk missile batteries, both of which appear on the US Munitions List as "defence articles".
Mr Tappin agreed to pay $5,000 apiece, which was $2,000 above the normal retail price, for 50 batteries, according to the court papers.
"During these negotiations, Tappin and the undercover agents discussed the specifics of the plot, including the use of double (i.e. false) invoices," it is claimed.
Mr Tappin told Mercury that Caldwell, "an Oregon fertilizer importer", would buy the batteries on his behalf and send them to him.
During Caldwell's case, prosecutors produced evidence that he had written down the words "batteries", "trucks" and "military" during a phone conversation with Mr Tappin.
"Caldwell testified and admitted that Tappin told him of the batteries' military application," the papers state.
Agents sent Caldwell two sets of invoices - a false set, disguising the nature of the batteries, and the true invoices, with a note saying: "Export of the commodities described herein is strictly prohibited without a valid export licence issued by the US State Department, Office of Defense Trade Controls, prescribed in the International Traffic in Arms Regulations."
Caldwell faxed the invoice to Mr Tappin with an asterisk and a note telling him "please see detail below", it is claimed.
Caldwell later told an undercover agent: "The thought of something that is potentially illegal isn't cool."
He then explained a plan to "disguise" the batteries before they were sent to Mr Tappin.
Caldwell was jailed for 20 months in 2007 after being convicted of charges relating to the sale.
Mr Tappin denies any wrongdoing, saying the batteries were destined for the Netherlands and were to be used for electroplating cars.
He claims his commission on the deal was a mere £320, but prosecutors say he stood to make £96,000 if other deals he set up had gone ahead.
They also claim to have recovered 16,000 documents from Robert Gibson, another co-defendant who was jailed for 24 months in 2007, which show that Mr Tappin had done business with Iran before.
Mr Tappin's British solicitor, Karen Todner, said: "If there is evidence against Mr Tappin we don't know about it because he was extradited without knowing the evidence against him. It's not for Mr Cameron to judge someone's innocence or guilt. If there is evidence against him or he is guilty, then let him stand trial in the UK."
