Kobori, 69, sits in a share of fourth at 12-under with Australian Min Woo Lee, the 2023 champion.
Hillier, 67, is another shot back alongside former Masters winner Adam Scott, who has won the event twice, fellow Aussie Marc Leishman and China’s Wenyi Ding.
Kiwi Nick Voke is also in the hunt at 10-under with his second straight 66, with Aussies Cameron Davis, Greg Ogilvy and Englishman Alex Fitzpatrick, brother of Ryder Cup winner Matt. Fourteen players are within three shots of the lead.
Kobori held the overnight lead after carding a 63 on Friday and had a lightning-quick start in the third round with birdies at the opening two holes.
However, players were called off the course because of the threat of lightning, after which Kobori couldn’t find another birdie, losing the lead.
Hillier looked set to be with Kobori just a shot off the pace before his only blemish, at the 18th, dropped him down the leaderboard. The Wellingtonian opened with three birdies on the front nine and added two more on the turn home before his bogey five at the last.
Fellow Kiwi and world No 41 Ryan Fox fell to 57th on the leaderboard after a round of 73.