Davis' team beat the Faroe Islands 3-1 away on Friday to top Group F on 16 points ahead of Romania (15) and Hungary (12).
"We are on the cusp of something really special. It would be huge and I would very proud to achieve that," the 30-year-old Southampton player was quoted as saying on his club's website.
"In my career it would be the biggest achievement to date," he added.
"It is such a difficult thing for a nation of our size to qualify for a major tournament and that's been shown in how far we've been off it in previous campaigns.
"It is hard not to get caught up in the expectation and hype surrounding this game," he said.
"There is a great prize there for us at the end of the game.
"Hopefully we can go out and hold our nerve and show the quality we have done up to now in the group."
(Reporting by Simon Jennings in Bengaluru; editing by Ken Ferris)
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