Ms Bibi has already spent nine years in jail after being accused of breaching the country's strict blasphemy laws by insulting the Prophet Muhammad, following a row with Muslim women.
She was sentenced to death by hanging in 2010.
If her appeal fails, she will become the first woman to be executed for blasphemy in Pakistan.
Wilson Chowdhry, chairman of the The British Pakistani Christian Association (BPCA), said in a statement: "The family of Asia have said that if Asia is exonerated they will flee to one of the many countries who have offered them asylum."
Ms Bibi's husband, Ashiq Masih and their eldest daughter Eisham spoke recently at an event organized by Aid to the Church in Need at the Catholic Chaplaincy of Lancaster University.
He said his wife would "never convert to Islam".
"She is psychologically, physically and spiritually strong," he added.
"Having a very strong faith, she is ready and willing to die for Christ."
There have been growing concerns about Ms Bibi's health.
Last month Norren Barkat, leader of Masih Overseas - a Pakistani-Christian humanitarian group in Switzerland, described what a journalist had told her about Ms Bibi's wellbeing.
"I am in touch with a Pakistani journalist who has met Asia and has described symptoms relating to her mental health such as memory loss and confusion, which I as an assistant nurse can clearly identify as early stages of dementia," she explained.
"Although she is not being treated for this in Pakistan and is simply being labelled as mad by prison staff."
Mr Chowdhry encouraged people to pray for the Christian woman's release.
"We hope this time she will be completely exonerated and these this wrongful conviction will finally be overturned as this is her final opportunity to be heard at court and freed via the judicial system," he said.
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