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Wayne Pivac and Dave Rennie have been linked to the role and the Guardian reports Crusaders coach Scott Robertson is also a contender.
Welsh Rugby Union chief executive Martin Phillips says the shortlist is more than two, and it's the same one they've had for a while.
"We will get the coach done this summer," Phillips told the Guardian. "The shortlist is more than two and it's the same one we've had for a while.
"We've been interviewing for a few years now and it's been a combination of doing that, having meetings, and going for dinner. We've still got some more of that to do and then we'll make the call."
Wales have been almost exclusively coached by New Zealanders since Sir Graham Henry took over in 1998.
Gatland has been in charge of the side since 2007, across more than 100 tests, an impressive achievement in the professional era.