Glover trailed Barnes by a stroke at the completion of the third round late Sunday but Barnes bogeyed the only fourth-round hole he played late Sunday while Glover notched a par.
Play will resume at Bethpage Black Monday morning.
Mickelson shot a 69 in Sunday's third round while Woods shot a 2-under 68, the low score of the day.
Barnes, who began the tournament ranked 519th in the world, had his toughest round after shooting 67 and 65 on the first two days. He eagled the par-5 fourth hole, somewhat offsetting five bogeys, four of which happened on the back nine.
Mickelson, meanwhile, had four birdies and two bogeys on the final nine holes of the third round. Woods played a steady round that included a bogey on 14 and birdies on 3, 4 and 17.