Thorpe did not play a senior game for Spurs, but made 18 appearances in Premier League Two in 2020-21.
The 20-year-old has moved to Kenilworth Road after turning down a new contract, having been allowed to train with Luton towards the end of last season.
"He's young and a very technically gifted midfield player," boss Nathan Jones told the Hatters website.
"I'm delighted to be here, it has been a long process but one that is definitely going to be worth the wait," added Thorpe.
He is the club's ninth signing since the end of last season but details about the length of his contract have not been disclosed.
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