But the odds that even one would survive the complex operation are extremely slim.
No twins born with one heart have ever been separated successfully.
Now doctors say that time is running out for their parents to make their choice.
Emma and Taylor, who are fused from the breastbone to the belly button, have already defied experts once.
They were expected to live for only 15 minutes because it was thought that their heart would become overworked as soon as they took their first breath.
But doctors say they must be separated in the next year or two to have any chance of survival.
The girls also share a liver. For the surgery to succeed, it is likely that they will both need heart transplants and one will need a new liver.
Their American mother Mandy Bailey, 32, who lives in Arizona, said: 'There's a tiny window of time for a separation. We have to get it completely right, if we wait too long then the operation cannot go ahead.
'In the next year, life for all of us will be drastically different.
'We know we have to make difficult choices. We understand the risks. But we also understand the outcome if we don't do anything.'
She said that when the girls were born, all her husband Tor, a 34-yearold builder, had hoped for was to hear them cry just once.
'We were told they would live for minutes and then they would suffocate,' she added. 'The girls were handed to him straight away and they simultaneously inhaled and quietly cried. It was a magical sound.'
The twins' future was still uncertain when they were well enough to be discharged from hospital. They were given oxygen and medication to strengthen their lungs and have astonished doctors with their progress.
They can feed themselves and are learning to get about using a special walker.
Mrs Bailey said: 'The girls are becoming different people and developing adorable little traits. Taylor is the bossy one.
'Emma is a bit crazy, she's always pulling funny faces.' The Baileys have four other children aged 11, nine, seven and two. Emma and Taylor have already had several smaller heart operations.
One in June last year involved placing a band around Emma's aorta to reduce her blood pressure.
Seattle Children's Hospital, which is overseeing the twins' care, does not want to discuss the treatment.
Mrs Bailey said: 'We realise what a great miracle it is for our daughters to be alive and happy. They are a blessing for our family.'
To find out more visit the Two Sisters One Heart Foundation (www.twosistersoneheart.com).
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