Vilips fired a six-under-par 66 and Davis a four-under 68 to be within striking distance of American Chris Kirk, who posted a blemish-free nine-under 63 in his quest for a seventh PGA title.
Kirk led the way after a gorgeous day on the southern tip of Baja California at El Cardonal at Diamante, a course designed by Tiger Woods.
The American was practically flawless on Thursday (local time), hitting each of the wide fairways and missing only one green.
Kirk is at No.110 in the FedEx Cup Fall standings with only three tournaments remaining for him to finish in the top 100 and keep a full card for 2027.
He is one shot ahead of compatriots Nick Dunlap and Zac Blair.
"Just go day by day, shot by shot. I know how cliche that sounds, but there's not really anything I can do to control where I finish on a points list," Kirk said.
"The only thing that I have somewhat of control over is how I approach each day and how I look at each shot."
Equally simple was explaining his round. He putted for birdie on every hole except the par-3 second, where he left his tee shot out to the right. He pitched that up to 10 feet and made par.
His worst swing was on the 156-yard 16th, where Kirk hit what he described as a "little bit of a chunk pull". He was left with a 70-foot putt over the ridge and holed it for an unlikely birdie.
"There was a lot to like, for sure," Kirk said.
"It felt like my swing was in a really good spot and I hit a bunch of really nice putts."
Scoring conditions were so pure that more than half of the 120-man field broke 70.
Vilips certainly had an eventful day, opening with an eagle on the par-5 first, then adding seven birdies, one bogey and a double-bogey as he carded a 66 to be just three shots behind Kirk.
Fellow Aussie Davis was two strokes further back, helping himself to four birdies to sit six behind the leader.
Ben James, who has a card locked up for next year as the No.1 player from the PGA Tour University ranking, was among those at 65.
Apart from Vilips, the large group of players at 66 included 19-year-old Blades Brown and Ryan Gerard, who was eligible for the European tour's post-season event in Abu Dhabi but chose not to travel to the Middle East.
- With AAP
