Wales captain Ramsey had taken temporary charge as Cardiff were relegated from the Championship last month, Omer Riza having been sacked with three games of the season remaining.
But while it is not clear if a final decision has been made, Barry-Murphy is now the frontrunner to land a first senior manager’s position since leaving Rochdale in 2021.
He would become the Bluebirds’ ninth manager in four years as they prepare to play in the third tier for the first time since 2003.
The Irishman has been on Cardiff’s radar for some time having been under consideration to replace Mick McCarthy in 2021.
Barry-Murphy is a former midfielder with Cork City, Preston North End, Sheffield Wednesday, Bury and Rochdale, becoming player-coach at the latter in 2014 at the age of 35.
He succeeded Keith Hill as manager in 2019, winning praise for the side’s style of play and work with homegrown talent despite the team’s relegation to League Two in 2021.
Barry-Murphy resigned from the job to take to take up the role at Manchester City’s academy, succeeding future Chelsea boss Enzo Maresca and winning two youth titles, coaching the likes of Cole Palmer, Oscar Bobb and Romeo Lavia while working under Pep Guardiola.
After quitting the Etihad three years later, Barry-Murphy went on to be named as part of Ruud van Nistelrooy’s backroom staff at Leicester in December 2024.
But Barry-Murphy had been keen for a frontline role of his own and could now be handed the challenge of returning Cardiff to the Championship.
Buckingham, who guided Oxford to League One promotion in May 2024, has also held talks with Cardiff.
The 40-year-old is currently without a club having left the Kassam Stadium in December following a poor run of results.
Buckingham, who previously coached in New Zealand and managed City Football Group clubs Melbourne City and Mumbai City, had previously been under consideration by the Football Association of Wales (FAW) before the appointment of Craig Bellamy as national boss.