The twins, Marieme and Ngeye, were born in Senegal three years ago.
Though they have separate hearts and minds, they are conjoined from the shoulder down.
The twins and their dad, Ibrahima Ndiaye, currently live in Cardiff, United Kingdom; while their mum reportedly went back to Senegal to attend to some family matters.
“The first thing you do is go to Google. There’s a one in 10,000 chance of this happening to your children and you think: why me?
“Your life, your career? You just press the standby button,” Ndiaye, told BBC 2.
The conjoined twins were not expected to live more than a few days after birth, but they had survived and are currently preparing to start full-time nursery, the BBC 2 reported.
The doctors and the parents have been unable to decide the best way forward as one of the girls needs urgent surgery to save a failing heart.
Marieme had developed a heart condition and the heart is failing.
Doctors worry that she would not survive the anaesthetic required for corrective cardiac surgery, or separation from her sister.
“The toxins her leaky valve pumped into their shared circulation would kill her twin sister, Ngeye,” doctors said.
They added that separating the twins meant that Marieme would die, but her twin sister, Ngeye, has a slim chance of survival in the aftermath, the physicians warned.
The parents and the physicians have been unable to arrive at a conclusion, it was reported.