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Published 16:43 5 May 2022 BST
Updated 17:03 5 May 2022 BST
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The former Ireland captain was on Monday Night Football with Jamie Carragher in April and expressed his desire to move into management.
"I've got all the badges, the pro license, I've been building up to this. I've done my apprentice as an assistant so I'm looking to get the right opportunity," Keane said on Sky Sports.
"I have had a few offers, but nothing that I really fancied. I'm sure the right one will come around the corner. If it does, I'll dive into it."
The 41-year-old could follow his former Ireland teammate, and current Shelbourne coach, Damien Duff into League of Ireland management if he was to take over at Shamrock Rovers.
Keane, if he achieves even a fraction of the success as a manager that he achieved as a player, could be the perfect successor to Bradley.
Rice has been the busiest of the three candidates since retiring from playing in 2016. The midfielder coached with Rovers Under-17s from 2016 to 2018 and the Under-19s side from 2018 to 2020 whilst doing his coaching badges.
The former Rovers skipper took up the opportunity to coach with Crystal Palace Under-23s ahead of the 2020/21 season. Rice remained in the role for 12 months before becoming Ireland's chief scout and opposition analyst.
Rice is arguably best placed to take over at Tallaght Stadium due to his playing and coaching experiences with the club. As McPhail remains the club's sporting director, their previous experience of working together should allow for talks to take place.
Compensation would be required to bring Rice back to the Hoops, however, it could be a match made in heaven.
However, the best option for Rovers is undoubtedly Bradley remaining at Tallaght Stadium.
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