The arrest comes just hours after the 23-year-old’s funeral.
The body of Ms Murphy, a talented musician and teacher, was found on the banks of the Grand Canal in Tullamore last Wednesday.
The man who has been arrested presented at the hospital late on Thursday with serious injuries, some of which were believed to be self-inflicted.
It is understood that he could not satisfactorily explain his injuries, with medics being concerned, leading them to alert Gardaí.
The man became a person of interest after checks into his background found links to Co Offaly. Subsequently, a house in a Co Offaly village linked to him and another in south Dublin were both sealed off and searched extensively.
Two vehicles, also linked to the man, have been taken for forensic examination.
He was arrested in Dublin on Tuesday and detained under Section Four of the Criminal Justice Act which allows Gardaí to question him for up to 24 hours without charge.
The suspect, who is aged in his 30s with young children, was arrested after he was deemed to be sufficiently recovered from his injuries.
A key line of inquiry for Gardaí will be to determine if the man was involved in the murder through cross-checking his DNA and samples taken from the crime scene.
Other forensic evidence investigators have include DNA and fingerprints from a Falcon Storm mountain bike abandoned at the crime scene as well as forensic evidence from Ms Murphy’s clothes and her remains.
Gardaí are also in possession of CCTV footage from the area on Wednesday, which appears to show a man cycling the Falcon Storm bike in the period before the murder.
It is understood that all the information gathered so far will be put to the suspect while he is being questioned by Gardaí. He will also be asked about his location and movements on the day of the murder and last Thursday before he was admitted to hospital.