The 17-year-old had been offered terms by Everton having been with the academy setup for five years after joining from West Bromwich Albion in 2016. But Small left as a free agent for Southampton last month.
Unsworth told the Liverpool Echo: "In fairness to the club, we did everything we possibly could [to keep him].
"We made him an unbelievable offer, we didn't want him to leave. We never envisaged it coming.
"Up until he actually did leave, we fought really hard to keep him. But when a player doesn't want to be at your club, it's very difficult to try and keep him.
"He's got his way, and that's usually the case these days."
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