The suspect is being currently treated at a hospital but is understood to be under ‘discreet surveillance’ by officers.
Sources say that he is not linked to the innocent man arrested in relation to the murder and later released without charge.
Gardai hope to speak him very soon, a source said.
Gardai eliminated an arrested man as a suspect in the Ashling Murphy murder investigation after fast-tracking evidence to Forensic Science Ireland (FSI).
Last night, the man was a placed in a safe house following his release from custody.
His solicitor Donal Farrelly told the Irish Independent that his client’s life has been made “unbearable” after he was identified online.
It is believed the arrested man was eliminated from Garda enquiries after a number of samples were sent to the FSI laboratory.
Given the gravity of the murder investigation, the samples were processed urgently and the man was cleared as a suspect.
It is understood gardai are now pursuing a "very good line of enquiry" and believe the case will be solved.
Today, the man's solicitor - Donal Farrelly, of Donal Farrelly & Co Solicitors - said that people can’t lose sight of what happened to Ashling Murphy but that his client was wrongly “vilified” by people.
“Doesn’t this show the vital importance of the presumption of innocence.
“This is the most important thing to come out of this, and we have to keep that in our focus.
“It is a terrible tragedy what happened to this lady, what happened to her and her family, but this man has been vilified on social media.
“Absolutely vilified, and everybody clearly identified who he was on social media.
“It has made life unbearable for him I would say.
“It is dreadful- the amount of hurt, and anger, and ire about him on social media is frightening,” Mr Farrelly said.
Donal Farrelly was speaking after gardai released a statement saying the man had been eliminated from their enquiries “and is no longer a suspect”.
In a statement, last night Gardaí said: “This male has been eliminated from Garda enquiries and is no longer a suspect.
The shock development comes after officers had earlier stated that they were only looking for one individual in relation to the attack.
Gardaí have now issued a fresh appeal for members of the public to come forward with information in relation the death of Ms Murphy.
It is understood that a post-mortem which took place yesterday showed she died by strangulation.
Two women witnessed part of the attack which took place along a popular jogging route in Tullamore
Detectives believe that other people out walking may have seen Ms Murphy or her attacker in the area around 4pm on Wednesday.
This morning, gardai renewed an appeal for information, in particular in relation to a Falcon Storm mountain bike described as having “straight handlebars and distinctive yellow/green front forks.
“Gardaí continue to appeal for any person with any information on this fatal assault, who may not already have spoken to investigators, to come forward with that information,” a statement said.
Anyone with information is asked to contact Tullamore Garda Station on 057 932 7600, the Garda Confidential Line on 1800 666 111, or any Garda Station.
Meanwhile, Ashling Murphy's heartbroken parents have paid tribute to their "beautiful, talented" girl as they attempted to come to terms with the devastation of her sudden and inexplicable death
Comforted by neighbours, friends, and the local community in the rural townland of Blueball near Tullamore, they spoke of a young woman with energy, drive, commitment, talent and love
The photographs on the walls of their family home reflect the love and pride that her parents have for all their children.
Now they are trying to think about what life will be like without their youngest child, their baby girl who was just starting out on her career as a teacher with a life of opportunities ahead of her.