Christine Peckham, 53, suffered two strokes and is partially blind and paralysed after taking the drug.
Documents relating to her case were sent to the US yesterday and are due to be filed with the courts there by American lawyers.
British solicitors acting for Mrs Peckham say the court case could start within 12 months. Russell Spargo, of the MSB firm of solicitors in Liverpool, said: "The documents have been prepared and need to be looked at by a cardiologist in the US before they are lodged with the courts. This is just the beginning of the process but we are prepared for a long haul."
More than 4,000 people, including 300 in the UK, plan to sue Merck for negligence over the best-selling painkiller. Merck has denied allegations that it covered up evidence that taking Vioxx was linked to an increased risk of heart attacks and strokes.
The company says it intends to contest every court case and has already set aside millions of dollars to pay for the lengthy and potentially costly legal battles.
But some experts believe that pressure is increasing on Merck to settle the class action after it lost the first case of its kind last week. A jury in Texas ordered the drugs company to pay £141m in damages to the widow of a man who died after taking Vioxx.
Merck faces three more American cases over the coming months, while Mrs Peckham may be the first foreigner to have her action come to trial in the US.
Michael Kelly, a US legal expert, told Bloomberg.com: "If they lose these next three cases, Merck will be in deep trouble. They will have given the plaintiffs a road map on how to successfully try these cases and the pressure for a global settlement of these suits may start to grow."
British people who claim they have suffered as a result of Vioxx are suing in the US because the Legal Services Commission refused them legal aid for a class action in Britain.
In the US system, the cases will proceed on a "no win, no fee" basis.
The alleged victims also stand to win much larger awards by going through the US litigation system.
Mr Spargo said: "In Britain, they may only get £30,000, but in the US some of these cases could be worth millions. This is not about winning large sums of money; for many people it is about getting an explanation and justice for what has happened to them."
He said hundreds more people had contacted his firm after the Texas award.
The UK drugs approval body, the Medicines and Healthcare Products Regulatory Agency, is investigating whether Merck withheld information about the potential risks of Vioxx when applying for a British licence in 1999.
The US court case heard that concerns about the link with cardiovascular events were raised by Merck's research unit as early as 1998. But as late as 2000, Merck was still aggressively marketing the drug as a safer and more effective alternative to older-style painkillers.
In that year, the company appealed against a decision by the National Institute for Clinical Excellence, the body which recommends drug use on the NHS, which had ruled that Vioxx should be prescribed only for people who could not take the older medicines for safety reasons.
Merck withdrew Vioxx last September after a new trial showed that taking it for more than 18 months could double the risk of heart attacks and strokes.