Manchester City were also keen on the 16-year-old midfielder, who is an England youth international.
But Liverpool have now moved to agree a deal reported to be worth around £250,000, although if he makes the first-team it could possibly rise to £1million.
Crewe spokesman Rob Wilson confirmed: "We are naturally disappointed that Dan has decided to sign for Liverpool.
"The club has received an undisclosed fee and future add-ons but we would have much preferred to have seen him play for our first team."
Smith, who has been with Crewe's highly acclaimed academy since the age of 11, also confirmed the deal when he Tweeted a picture of himself with Liverpool boss Kenny Dalglish.
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