After missing numerous opportunities throughout his first two rounds, a crucial birdie on the 13th kept Northern Irishman McIlroy in the mix.
A shocking run of three bogeys in four holes did not derail Kuchar’s bid as the American recovered to finish Saturday with a one-under round of 70.
Meanwhile, Australian Scott holed six birdies to storm into contention with a four-under 67.
The trio are closely followed by Dustin Johnson, who showed some of the form that saw him clinch the Riviera title in 2017 as he ended the penultimate round birdie-eagle-par for a 67.
McIlroy is staying focused ahead of Sunday’s final round, with the three leaders on 10-under and just four shots separating the top 16 players.
“You just have to worry about yourself, concentrate on what you’re doing, do it well, set yourself a target, don’t think about anyone else and if that’s good enough at the end of the day, then great,” he told the PGA Tour website.
“If not, then someone just played better than you and hats off to them.”
Kuchar is keen to add the title to his list of trophies.
He said: “There’s not a PGA Tour event you wouldn’t want to win, but there are a handful that are extra special, have a great list of past champions, have a great golf course. This one’s one of those.
“You chalk it up as one of those sort of extra special Tour events that you really hope to put your name as a past champion.”