The Australian will enter the final round in fourth spot, three shots shy of leader Colin Morikawa on a baked-out Bay Hill course in Orlando.
"It's going to be a lot more difficult day tomorrow," said Day, chasing his first PGA Tour win since the AT&T Byron Nelson in May 2023.
"There's going to be 10 to 20 mile an hour wind out there.
"We all know how hard the greens are and how hard the golf course is, so it's just kind of survive, get close to it on the back side.
"You just got to try and cut into the lead through the first nine - if I can cut into the lead, get myself on the back side, hopefully finish the way I did today, then you never know what happens."
Day finished his third round with three consecutive birdies to post a three-under 69 - he's in the mix after following his opening 76 with a tournament-low 64 on Friday.
The 37-year-old former world No.1 briefly tied for the lead but closed out his front nine with three straight bogeys on a course hardened by three days of sunshine.
Some golfers watched birdie putts from a reasonable range roll past the hole by eight feet or more.
"Any time you put your putter down and you start sliding on your putter, that's when you know the greens are getting pretty slick," Day said.
"They're changing colour and you can kind of start to see the change in colour."
Morikawa made a 12-foot birdie putt on the 18th for a five-under 67, giving him a chance to win for the first time in 17 months.
"You have to hit good shots out there," said Morikawa, who is at 10 under.
"It can get you from any direction."
Fellow American Russell Henley holed a long bunker shot for birdie on the opening hole, ran off four in a row to start the back nine and had a 67 to finish one shot behind. Corey Conners of Canada carded a 69 and is two back.
Irishman Shane Lowry, the 36-hole leader, had a 76 to leave him six shots behind.
Rory McIlroy (73) made bogey on three of his last four holes to fall seven shots adrift, while Scottie Scheffler could only manage a 71 to be eight shots back.
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