Kelly moved to Everton a year ago from Preston North End.
The Everton goalkeeper coach, Alan Kelly, is a target for Mick McCarthy, according to the Irish Sun.
McCarthy has just began his second spell in charge of the Republic of Ireland national team, and is said to want Kelly to return to the role he also occupied once before, between 2006 and 2013.
The tabloid reports that the former Ipswich Town boss hopes to strike a deal with his Everton counterpart, Marco Silva, which would allow the 50-year-old to combine both roles.
Kelly, who won 34 caps for Ireland between 1993 and 2002, moved to Everton late last year when David Unsworth was in temporary charge at Goodison Park. The pair had previously worked together at Preston North End.
Everton players Seamus Coleman and James McCarthy are regulars in the Irish set up, while Harry Charsley has been capped at Under-21 level.
Should Everton free Kelly for Ireland duty?
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