Vincent Hickey, 41, was treated in the hospital wing of Long Lartin prison, near Evesham, Worcestershire, after slashing his wrists on Thursday.
He had been found covered in blood in his cell.
A Prison Service spokesman said the attempt appeared to be "a determined effort". Although the injuries were serious, they were not life-threatening.
Hickey, his 33-year-old cousin Michael, and James Robinson, 61, are serving life sentences for the murder of 13-year-old Carl in 1978 in Stourbridge, West Midlands.
The three have protested their innocence since their conviction and supporters and relatives claim there is now fresh evidence which could cast doubt on the conviction.
But Home Office ministers made it clear on Wednesday that, although they would look at any new evidence, the feeling was that the case would not be referred back to appeal.
Hickey's solicitor, James Nichol, said he could not comment on the suicide attempt until he had further details.
He said the prisoners' legal team had given the Home Office 28 days to respond to new evidence or solicitors would seek a judicial review.