The man, aged in his 30s, is being detained by gardaí and questioned in relation to potentially withholding information.
It follows the arrest of a main suspect yesterday in connection with the killing, who also remains in garda custody.
Sources said that while gardaí would have preferred not to have made the arrest on the day of Ms Murphy’s funeral, they felt they had to for risk of losing track of the suspect, after he was released from hospital.
Ms Murphy was murdered as she jogged along the banks of the Grand Canal in Tullamore, Co Offaly, a week ago.
The suspect was taken into garda custody under Section 4 of the Criminal Justice Act 1984 and questioned on suspicion of the murder of the 23-year-old teacher.
Gardaí have been monitoring his status at the hospital after he presented himself there last Thursday with multiple wounds.
Garda sources said they have a “huge amount” of evidence to put to the man during his various interviews in custody and they will need the time allowed under the act, unless anything the suspect says changes that.
Gardaí received a good response to the appeal on Monday for information about sightings of a man wearing a black tracksuit. It is thought an appeal, issued at the same time, asking if anyone had given him a lift, was a bid to ‘flush out’ those who did and give them an opportunity to come forward to gardaí voluntarily.
Examinations will take place to determine the nature of the suspect’s wounds and what could have caused them.
It is understood gardaí have already secured samples of his DNA from the hospital, on foot of a court warrant. They are also thought to have possession of clothing and medical material linked to him.
At the moment, gardaí have nothing to suggest there was any connection between the man and Ms Murphy, but detectives will question him about this.
Gardaí are liaising with foreign police forces in a bid to gather any international records in relation to the man.