All three Bobs shot 2-under 69 at the TPC at Avenal. May and Estes stand at 10-under-par 203, while May is tied with Jay Williamson (67) at minus 9.
Four players share fifth place, including 1999 Kemper champion Rich Beem (69). First-round leader Franklin Langham, Duffy Waldorf and Andrew Magee all shot 70 to finish at 8-under.
Overnight leader Greg Norman faltered with a 3-over 74 in round three. Hes still in contention, however, tied with four, three back.
The 47-year-old Norman started the day with a two-stroke advantage at 10-under, but was caught early by Burns. And despite bogeying the par-4 seventh to fall to 9-under, Burns managed to take the outright lead when Norman found trouble one hole behind.
Going for the green in two at the par-5 sixth, Norman pulled a 5-iron from 221 yards into the creek short of the green. Following a penalty drop some 60 yards from the hole, he chunked his fourth shot and made a double-bogey-7.
Norman struggled to make pars over his next three holes. Upon making the turn, he pulled his tee shot at the par-4 10th into the left rough. He tried to muscle his approach out of the thick grass, but it came out heavy and bounded off a stretch of rocks. Though his ball stayed dried, it took two more swings to find the putting surface, from where he one-putted for bogey.
Three-over on the day, Norman was still only two off the lead, as Burns bogeyed the 12th to drop to 9-under. Norman quickly cut his deficit to just one by rolling in a 12-footer for birdie at the par-3 11th. He then played his final seven holes in 1-over.
'It was a really strange day,' said Norman. 'Nothing really happened. Hopefully, I've got my bad round out of me, and I'll be back in the 60s tomorrow.'
May made his run up the leaderboard with birdies on 14 and 15. He briefly stood alone at 10-under, until Burns hit the green in two at the par-5 13th and two-putted for birdie.
Estes made it a trio of Bobs at the top by also birdieing No. 13. He then added another birdie at the par-4 16th to move one clear at 11-under.
All three men missed the green at the 195-yard, par-3 17th, with Burns, who snaked in a 10-footer, the only one of the three to save par.
Estes is the only one of the Bobs to have won on the PGA Tour. He took home two titles a year ago. However, this event often rewards the first-timer. Each of the last three winners has made this event his maiden tour victory.
'That's neat,' Burns said of the recent trend. 'Hopefully that trend continues.'
Full-field scores from the Kemper Insurance Open