Fox carded seven birdies and one bogey in his second round at the talent-packed Irish tournament; an event where the top three players who are not already exempt can book their ticket to the Open.
Lee is also within touching distance of a spot at the year's third major, where he could make his third consecutive appearance.
Lee strung together four birdies and a double bogey at the Greenbrier to post a two-under 68, moving to eight-under, four shots off the lead held by Sebastian Munoz.
The defending champion of the event, Lee explained why he's found success at the West Virginia tournament.
"I think the layouts here - it suits very well for my eyes. I don't have to hit a lot of drivers out here, I love hitting three woods off the tee and some of the courses, it's too long for me to do that," said Lee.
"I just keep giving myself birdie opportunities out there, so that's working really well for me."
The Greenbrier Classic offers four Open berths to the top four finishers who are not otherwise exempt, giving Lee two goals to chase in his final two rounds.
If Fox and Lee can book their ticket to the Open Championship, it would mark the first time three Kiwis played in a major since the 2009 Open, where Josh Geary, Mark Brown, David Smail and Michael Campbell all took part.