As a result, a strong final round should give Fox a chance to play in his second major, following his tie for 49th at the 2015 Open.
Things are slightly more difficult for Lee, who dropped back into a share of eighth at the Greenbrier Classic on the PGA Tour.
Lee, who is the defending champion at the Greenbrier event, had two birdies with two bogies to card an even par 70, and is six shots behind leader Sebastian Munoz.
After hitting 12 fairways in both his first two rounds, Lee lost his radar in his third round, hitting just seven. However, he continued to hit excellent approach shots, and managed to cobble together an average round to stay in touch.
The Greenbrier offers four Open berths to the top four finishers who are not otherwise exempt, and Lee is currently three shots outside the cut-off mark.
If unsuccessful, both Fox and Lee will have one final chance to qualify, with next week's Scottish Open providing three qualifying spots, and the John Deere Classic offering one.
Michael Hendry is currently the sole Kiwi to have qualified for the Open, which tees off in two weeks.