Putnam, who trailed by one stroke at the halfway stage, shot a three-under 67 to get to 16 under overall.
Kuchar has been one of the most consistent players on the Tour over the past few years, finishing in the top 20 in FedEx Cup standings in 2017, 2016 and 2015.
Anirban Lahiri played extremely well in the second round and yet came back only with a two-under 68 for the second successive day to be placed Tied 35th at the Sony Open in Hawaii. Putnam birdied No. 18.
Kuchar, who had been winless for almost four years until his win at the Mayakoba Classic late in 2018 now appears likely to add a 9th PGA Tour title just three months after his 8th.
Putnam, who won for the first time previous year in the Barracuda Championship held opposite a World Golf Championship, did his best to stay in reach.
Matt Kuchar made it look easy Saturday in the Sony Open. He excelled with the irons again Saturday, leaving himself birdie putts of reasonable length.
Kuchar was flawless on day three, posting four birdies without dropping a shot, to be the 54-hole leader at 18 under. He had a shot for another birdie on No. 6, but his chip from the fringe sat just outside the cup. That would have made for birdies on three consecutive holes.
"It was awfully steady, bogey free, the kind of golf I like to play, give yourself enough opportunities, enough will go in".
The strangest round belonged to Reavie, who made more eagles than birdies in his round of 65. 11-13 and regrouped for two birdies the rest of the way.
Bryson DeChambeau made a move with 63, tying his best score on the PGA Tour.
The 25-year-old won the Memorial Tournament, the Northern Trust and the Dell Technologies Championship in 2018, and moved up 30 places after mixing just one bogey with eight birdies, including three in a row on the way back to the clubhouse.
He's among a cluster of golfers at 11-under.
Also tied for fifth were Charles Howell III and 54-year-old Davis Love III, who had one of his better putting rounds. They all checked in with 64s.