He's aiming to snatch the deal from the French, who are pushing Dassault Aviation's Rafale multirole jet, and hoping to exploit a wave of pro-French fervor in the Arab monarchies of the Persian Gulf after Paris torpedoed a U.S.-Iranian rapprochement that these states bitterly oppose.
In the current political climate Cameron is likely to find himself with a dogfight on his hands, particularly since Europe's defense industry must rely on arms exports in these days of shrinking domestic budgets.
But he may find he's got a big gun or two on his side -- like Queen Elizabeth II, a frequent visitor to the gulf and who in April hosted Sheik Khalifa bin Zayed al-Nahyan, the Emirates president and the ruler of Abu Dhabi, the oil-rich economic powerhouse of the seven-state federation.
The queen took the Arab potentate for a tour of her Windsor Castle estate outside London in her horse-drawn royal carriage.
"It's no coincidence then that since April executives within the Eurofighters consortium of the U.K.'s BAE Systems, Pan-European EADS and Italy's Finmeccanica have privately grown far more confident about Typhoon beating its French rival ... in the race to help Abu Dhabi double its air force," observed British defense analyst Carola Hoyos.
Abu Dhabi, the Emirates capital, runs the federation's armed forces and military procurement program.
"In May, the Queen invited King Hamad al-Khalifa of Bahrain to watch equestrian endurance racing at Windsor with her, despite the outcry of human rights activists" against Bahrain's alleged abuses against opponents of the royal family, Hoyos noted.
"Three months later Typhoon added Bahrain to its expanding list of suitors from the region."
Meanwhile BAE is negotiating with Saudi Arabia on the pricing of 72 Typhoons the kingdom bought under the al-Salaam contract signed in August 2006 with an expected value of $16.2 billion.
It's also trying to persuade Riyadh to buy 72 more new-model Typhoons even though Riyadh's already buying 84 Boeing F-15S jets as part of a 2010 U.S. arms package worth $57 billion, with upgrades for older F-15S variants it already possesses.
In November 2012, Cameron toured the gulf and secured a $4.04 billion order for 12 Typhoons and eight Hawk jet trainers from the sultanate of Oman, and set up a defense industrial partnership with the Emirates.
British officials say he also secured a commitment from Abu Dhabi to consider buying as many as 60 Typhoons.
That was seen as a breakthrough because Paris had been lobbying hard for years to sell the Rafale to the Emirates.
Indeed, in 2009, then-French President Nicolas Sarkozy, an enthusiastic champion of French arms exports, even set up a naval air base in the Emirates to encourage it to buy Rafale, which has largely relied on sales to the French air force and navy.
But the negotiations hit a snag when Abu Dhabi wanted a more powerful engine, and a reduced price per aircraft. That gave BAE an opening.
"Cameron worked the phones and pressed the flesh in a way that even his detractors see as going beyond the call of duty," observed Francis Tusa, editor of the British publication Defense Analysis.
He noted with Typhoon orders from Saudi Arabia, Oman and Bahrain showing interest, BAE said it believes scoring in the Emirates would open up sales in Kuwait and Qatar.
If they pulled that off it would mean a clean sweep for Typhoon with all six member states of the Gulf Cooperation Council.
So there's a lot riding on the Emirates contract.
"For France and Dassault, a loss in the Emirates would be a bitter setback," Hoyos noted.
"Rafale relies on France for its survival, but Paris can no longer afford alone to shoulder the $2.02 billion-$2.7 billion annual cost of keeping up production and has halved its order.
"Ending Rafale's production line would maim France's military aerospace industry and undermine the diplomatic and military influence it gains from being one of the few countries able to rely on entirely on its own equipment."