Scott, who squandered a four-shot lead with four holes to play at Lytham in 2012, was one of just 13 players in the second half of the field to break par as the wind picked up in the afternoon.
A four-under-par 68 left the former Masters champion just two behind McIlroy, who returned a flawless 66 as the early starters made use of ideal conditions.
"It's important, you've got to keep pace with a guy like Rory," Scott said. "He has the potential to really put his foot down. I've seen him shoot lots of low rounds and there's nothing stopping him from doing that tomorrow, either.
"I don't want him running away [with it]. We've seen what happens, he wins majors by eight shots [in the 2011 US Open and 2012 US PGA]. And Martin Kaymer did it just a few weeks ago in the US Open and it makes for a tough weekend when you're that far behind having to force it."
Scott bounced back from his Lytham nightmare to win a green jacket at Augusta last April, but admitted getting his hands on the Claret Jug remained a massive goal.
"This has been a big focus for everyone in my crew this year," added Scott, who also led on the back nine of the final round at Muirfield last year before finishing third.
"We've had our minds obviously on the other majors, but we felt we've been close the last couple of years in the Open and obviously to win the Claret Jug is such a huge thing. Our mind has been firmly set on Hoylake this year and doing well here."