Carney's death last Monday, after boarding a bus from Killarney to Dublin before taking a last-minute flight to Istanbul.

5 Al-Saqar appears with Carney and her 13-year-old daughter in an undated photo before her murder July 7. Facebook/Jamey Carney

Police in Ireland suspect he is now in hiding in remote tribal areas of Jordan, where he is believed to have contacts, after traveling through Syria, the Irish Independent reported.

Cops now believe Carney was beaten with a heavy toilet cistern lid before she was suffocated with a hand over her mouth as she lay in bed, the Irish Sun reported.

5 Carney died of suffocation after neighbors reported hearing an argument coming from the rented home. Instagram / @carney_jamey

He had become needy and possessive in recent months, leading to Carney to decide to end the relationship, the outlet reported, citing police sources.

Just one week before the murder, Carney confided in a friend that Al-Saqar had asked her for the equivalent of $5,700.

Her bloodied body was found under the blankets of her rented home by her 13-year-old daughter on Tuesday, who called a family friend who alerted the police.

5 Al-Saqar, who was in the process of seeking asylum in Ireland, is believed to have fled to the Mideast throughTurkey. instagram/carney_jamey

The official cause of death was given as asphyxiation.

 She was beaten with the heavy lid of a toilet cistern and her body concealed under a blanket in her bedroom. It's a shocking way to take anyone's life,  a police source told the Irish Sun.

US authorities are believed to be working with Irish police and Europol to confirm Al-Saqar's whereabouts.

However, even though the US has an extradition treaty with Jordan, such extraditions have been rare.

The US National Central Bureau, the US's direct liaison with Interpol, made up of personnel from both the Department of Justice and the Department of Homeland Security, is said to be monitoring the investigation.

Carney's family in New York is also reportedly set to ask the US to support the Irish authorities.

Her mother and sister have flown to Ireland to support her daughter, after launching a GoFundMe to cover Carney's funeral costs.

The appeal, which organizers say will allow Carney's daughter to  stay in the land she loves, especially while grieving this tragic loss,  has raised more than $20,000 as of Monday.