The loggerhead turtle, known as Terri, was found on a beach in Jersey 12 days ago, and has since been treated by the New Era Veterinary Hospital.
Terri is currently housed in a special water tank at a fish suppliers, but vets say she needs a warmer climate.
Vet Peter Haworth, who launched the campaign, said: "She wants to go free".
So far more than £2,500 has been raised by the funding campaign website, but Mr Haworth said they need to hit the total within the next week.
He said: "We're not stupid...it's a huge huge amount of money but we were given a very ill animal to try and save."
The campaign group Get Terri the Turtle Back Home said there were no direct flights to the Canary Islands until late February, by which point the turtle might have died.
The group said chartering a private jet at a cost of £22,000 was "the most reliable way to ensure her a future", and no-one was profiting from the campaign.
The turtle's medication expenses have been covered by the New Era Veterinary hospital, while her water temperature has been regulated by The Fresh Fish Company.
Mr Haworth said Terri was now "very strong" but "the water depth we can have her in here isn't deep enough to encourage her to eat".
Mr Haworth said if the target was not reached they would try to send the turtle to an aquarium in the UK.

Maria Pridmore, 36, was jailed for three years in August after she admitted a string of frauds.
They included obtaining more than £16,000 in housing benefit, council tax benefit and income support.
Pridmore, who was described in court as a "Walter Mitty" fraudster, also stole over £30,000 from four women.
She appeared by video-link at Lincoln Crown Court from Drake Hall prison in Staffordshire where she is serving her sentence.
The court heard Pridmore, a former drug addict, benefitted by £56,679 from her crimes but had no available assets which could be confiscated to pay back her victims.
Judge Simon Hirst ruled that in the circumstances a nominal sum of £1 should be paid.
Pridmore made news in 2012 when she gave birth to a daughter after previously suffering 14 miscarriages, a stillbirth and the death of an infant son.
At her sentence hearing, the court heard the former nurse had been claiming to be a lone-parent with dependent children in order to get benefits, but at various times failed to declare she had moved or was living with her then partner as man and wife.
Pridmore also pocketed more than £30,000 from friends and two elderly women, and used the bank details of one of her victims to pay for a £60 taxi fare to attend a court hearing.
She stole almost £8,000 from one victim, an 84-year-old vulnerable woman.
Pridmore, of Angel Lane, Holbeach, Lincolnshire, previously admitted seven charges of benefit fraud between August 2010 and March 2013, 19 charges of fraud and theft between January 2013 and May 2016 and stealing her father's flat-packed kitchen in April 2015.

Jordan Maguire was stabbed at his house in Thrales Close, Marsh Farm, in May 2013 and later died in hospital.
Jason Nelson, 32, was detained in Trinidad in October 2013. He was extradited to the UK and appeared at Luton Magistrates' court on Saturday.
Mr Nelson was remanded in custody ahead of a crown court appearance later this month, Bedfordshire Police said.

Organisers of the event, which has seen Judi Dench, Stephen Fry, BB King and JK Rowling appear, want to "secure its future for the next 50 years".
They say each year they must raise £500,000 to keep the event afloat.
Prince Charles said the festivals "played a crucial role in nurturing new talent."
The festivals' Future 50 Appeal aims to raise £1m for the arts charity "that has been at the heart of Harrogate for half a century - to secure its future for the next 50 years", organisers said.
Backing the appeal, Prince Charles said: "I am so pleased to be able to support such an innovative charity organisation as it launches the Future 50 Campaign, which will help safeguard the festivals' heritage and reputation for excellence, as well as playing a crucial role in nurturing new talent and providing opportunities for all, making life that little bit more worthwhile."
Sharon Canavar, Head of Harrogate International Festivals, said: "We have a strong heritage and an ambitious future. We are passionate about building on this remarkable legacy for future generations.
An endangered turtle will fly by private jet to the Canary Islands at a cost of £22,000, if an online fundraising campaign succeeds.
[NEXT_CONCEPT]
A woman who pocketed over £50,000 by claiming false benefits and targeting vulnerable people has been ordered to pay back a token amount of £1.
[NEXT_CONCEPT]
A man has appeared in court charged with the 2013 murder of a 20-year-old at his home in Luton.
[NEXT_CONCEPT]
The Prince of Wales has given his backing to a campaign to raise £1m to secure the future of Harrogate International Festivals.