Welsh Rugby Union chief executive Martin Phillips says they are closing in on a decision.
"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," Phillips said.
Wales have had a strong New Zealand coaching influence since Sir Graham Henry took over in 1998.
From 2002-2004, now All Blacks coach Steve Hansen was in charge, while Gatland has been in charge of the side since 2007, across more than 100 tests, an impressive achievement in the professional era.
It's not the first time this year that the Crusaders coach has seen his name appear in reports of possible successors for an international role.
Last month, reports indicated Robertson was being scouted as a potential replacement for Wallabies head coach Michael Cheika, who will hand over the reins if his side fail at next year's World Cup.
Since taking on his first head coaching role with Canterbury in 2013, Robertson has continuously seen success. In four seasons with the provincial side, Robertson won three Mitre 10 Cup titles. In 2015 he coached the New Zealand under-20s side to a World Championship, and in his debut season at the helm of the Crusaders in 2017 he won the Super Rugby title.
This season the Crusaders were on track for a long run in the playoffs and perhaps a chance to defend their title, losing just two of their opening 14 matches.
With just two regular season games left on their Super Rugby schedule, the southerners look set to finish at the top of both the New Zealand conference and the overall ladder.
With Super Rugby on hold due to the international window, the Crusaders will play a friendly against the French Barbarians in Christchurch on Friday night, with six players set to make their Crusaders debut.
Crusaders: Mitchell Hunt, Israel Dagg, Tima Faingaanuku, Rameka Poihipi*, Jone Macilai, Mike Delany, Mitchell Drummond (c), Ethan Blackadder, Billy Harmon, Mitchell Dunshea, Hamish Dalzell*, Quinten Strange, Michael Alaalatoa, Andrew Makalio, Donald Brighouse. Reserves: Seb Siataga*, Ryan Coxon*, Chris King, Luke Romano, Whetu Douglas, Jack Stratton, Sam Beard*, Caleb Makene*
* - on debut.