Davis holds the record with six titles, a tally the 41-year-old is on course to equal, while he is also hoping to draw level with Stephen Hendry's 18 'Triple Crown' triumphs.
O'Sullivan move into the third round in York after beating world number 73 Michael Georgiou 6-1, which included breaks of 69 and 57, but played down talks afterwards of equalling records.
"I just want to play for the enjoyment of it," O'Sullivan said after the win. "I find it tough sometimes, I just have to manage it as well as I can.
"Records are not something I am worried about. I keep it as business like as I can - I don't want to get too emotionally involved."
O'Sullivan will face Michael White in the next round on Tuesday after the Welshman beat Matthew Selt 6-1.
"I don't rest on my laurels," O'Sullivan added. "What's the point of thinking like that for the next five, six months? That's a real loser's attitude.
"My game has started to feel good again and I thought I owe it to myself to give it a good go."