World number 11 Kim joined Webb, who carded a 70, on six under par after a birdie on the 18th completed a 69.
Webb, 42, is seeking an eighth major title, with her last coming in 2006, while Kim, 24, is aiming for a first.
Overnight leader Christie Kerr dropped to third place, two shots back, after a round of 73.
England's Georgia Hall is the best-placed Briton on level par, six shots adrift, with compatriot Bronte Law and Scot Carly Booth 10 shots behind on four over. Charley Hull plummeted to 12 over after a third-round 80.
Booth, one of three Scots who made the cut, carded a 75 to go with earlier rounds of 75 and 70, after a painful triple bogey on the 15th hole.
"The weather was pretty brutal to be honest," said the 2012 winner. "I played OK today apart from my four-putt from five feet.
"I had a triple on the 15th which was just stupid but apart from that it was good golf.
"I rushed the putt and my head went a little. It's just one of those; you have to forget about it now.
"I'll kick myself for a little bit, but tomorrow is a new day."
Catriona Matthew, looking for a third Scottish Open victory, posted a 76 to finish on eight over, while Sally Watson carded a 77 for 10 over.
Australian Karrie Webb and South Korean Kim Sei-young share the lead heading into the final round of the Ladies Scottish Open at Dundonald Links.