Ex-Sunderland manager favourite to replace Martin O'Neill as Republic of Ireland boss
O'Neill and his assistant Roy Keane parted company with the Republic on Wednesday after over five years in their roles.
The Green Army finished bottom of Group B4 in the UEFA Nations League and had won just one of their past 11 games.
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Hide AdO'Neill, who managed Sunderland between 2011 and 2013, took charge of the Irish national side back in 2013, before guiding them to the knockout stages of the 2016 European Championships.
However, Monday's goalless draw against Denmark proved to be his last game in charge.
FAI chief executive John Delaney said: "I would like to thank Martin, Roy, and the management team for the impact that they had with the Ireland team.
"There have been many highlights during Martin's reign - none more so than Euro 2016 in France, which will live long in the memory of all Irish supporters."
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Hide AdMcCarthy, who led Ireland to the 2002 World Cup before taking charge of Sunderland, is now the favourite to replace O'Neill.
The former Ipswich Town boss, who has not returned to management since leaving Portman Road, is the clear bookmakers' favourite, with Paddy Power offering odds of 5/6 for him to get the job.
Former Black Cats managers Sam Allardyce and Steve Bruce are also among the front runners at 10/1, with both currently out of work.