Furue held a three-stroke lead when play was suspended due to dangerous weather conditions on Friday at the fourth of the season's five women's majors.
She created lear birdie opportunities on the 15th and 16th but failed to capitalise as she settled for a second successive round of 65 to take her 12 under at halfway.
The Japanese player has been a model of consistency this season securing eight top-10 finishes in 16 tournaments.
The Evian Championship leader has carried that form into this tournament with 12 birdies, 24 pars and no bogeys over the first 36 holes.
Her was lead trimmed when play resumed on Saturday by Kyriacou (66) and Tavatanakit (68) who made the most of a sunny morning.
Hye-Jin Choi (68) also stormed back into contention, following her two bogeys on Friday with four birdies on Saturday morning to leave her three shots off the pace.
The third round tees off early Saturday afternoon.
