The club take on Sunderland today (12.45pm) and can earn a place in the quarter-finals with a victory at the Stadium of Light.
With a number of big names exiting the competition, there is a sense that the door could really open for Saints if they can progress this afternoon.
“We can dream of doing something special in the competition, but you never want to say too much regarding that, because, obviously, you’ve got difficult fixtures ahead of you,” said Davis.
“We’ll take one game at a time, starting with Sunderland. “There’s a real desire to go on and do well in the competition.
“If we can get that little bit of luck, perform on the day, and show the quality we have got, then there’s no reason why we can’t go on and do well.”
Davis has never advanced beyond the fifth round of the FA Cup, but he has enjoyed greater success in the Scottish equivalent, winning it twice with Rangers, while also picking up three League Cup titles north of the border.
“I think any silverware that you’re fortunate enough to lift stands out in your mind,” said the midfielder.
“There’s nothing better than lifting a trophy, no better feeling in the world.
“As I say, the desire’s there to go on and do well in the cup.
“We’re desperate to do well, and we dream of doing well in the competition, but, obviously, there’s still going to be difficult fixtures ahead of us.”
Saints are currently enjoying an excellent run of form, with Tuesday’s 1-0 win at Hull pushing them up to eighth in the Premier League and extending their unbeaten streak, across all competitions, to eight matches.
Davis believes the team are yet to reach their peak, however.
“I think there’s a lot of quality, and one of the best things about this team at the minute is that there’s still a lot of progression within the team, and we feel we can improve a lot more,” he said.
“I think we’ve shown that, even from the start of the season.
“Since the manager’s come in, we’ve shown a consistent level of improvement and progression, and I still think there’s a lot more to come from this side.”
A host of players have made major contributions to Saints’ success this term, but Davis is one who has arguably not received the focus he deserves.
The 29-year-old has started the club’s last 13 league matches and his influence has become increasingly noticeable.
“I think when I first came (to the club) I was really frustrated that I wasn’t hitting the levels that I knew I was capable of, and I think it’s more of a case of just getting back to the level that I know I can play at, and doing that consistently,” he said.
“Obviously, when I first came, I came off the back of an operation and missed pre-season and that took me a long time to get over as well.
“I feel, sort of since probably at least the last year, I’ve maybe been getting back to the level that I’ve shown at, but I still know there’s a lot more in me and I’m capable of more as well.”