Thompson, who played 28 holes on Saturday after several players' second rounds were delayed by wind, finished two shots clear of Norway's Suzann Pettersen at 13-under par.
The American, whose only major win came at the event in 2014, holed six birdies and a bogey in her third round at the Mission Hills Country Club in Rancho Mirage, California.
Thompson was the big mover, finishing off a second round of 67 early in the day before doubling the dose with a third round 67 to graft a two shot lead at the top of the leaderboard.
Lee is tied with three South Koreans, Park (68), Mi Jung Hur (65) and Ryu (69).
Michelle Wie and Cristie Kerr of the United States, world number two Ariya Jutanugarn of Thailand and France's Karine Icher are five shots behind Thompson at eight-under.
She's tied 36th at one-under-par. "It's been kind of the story of the last few days". "My eyes are more over the ball".
"I just love this golf course", Thompson said.
"I'm just trying to focus on my own game, so you can't really let that get into your mind".
The 21-year-old, from Kettering, finished second last year after champion Lydia Ko birdied the final hole, but a repeat looks unlikely as she sits eight shots off the lead with 18 holes to play.
Wie had back-to-back birdies at the second and third but couldn't build on that.
Calm Sunday...Severe Weather Threat Monday
Moisture levels will increase significantly after the warm front moves through and that is when we are most likely to see storms. Midweek, the next low pressure system will move into the Plains and Midwest, possibly bringing the risk of severe weather.
Pettersen is determined not to let the ANA slip away this time.
She said a visit with the physio and a good night's rest would set her up for Sunday.
A quartet of players were in the clubhouse one shot behind Pettersen. "I feel like I was swinging at it a little exhausted coming in". "But I'm okay, just hung in there today".
"I had a lot of chances". Their younger brother, 16-year-old Sebastian, is a promising teenage tennis player who just reached the semifinals of the Easter Bowl USTA Junior National Spring Championship just down the road at the Indian Wells Tennis Garden.
Wie, whose career has been plagued by injuries, has been playing with a renewed enthusiasm and a new putting stance that is working very well, lifting her spirits for a final-round charge. "I'll be good", she vowed. The field got all caught up on a flawless desert day.
The third round begins at 11:30 a.m. local (Pacific) time with players going off in threesomes.
The greatest threat to Thompson could be Inbee Park, who has 18 LPGA wins, including here in 2013, a year in which she captured the first three majors of the year.
Yes, she won a U.S. Open, and she's won LPGA events in her native Hawaii, in Canada and in Mexico.
Pettersen said the switch was all about logistics. "It doesn't matter how many majors I have".
Meanwhile, 14-year-old American amateur Lucy Li is above the projected cut line at one under through eight holes.