The winger, 19, was out of contract at White Hart Lane this summer and had already been told he was to be released by the Premier League club.
He arrived at the Abbey Business Stadium on loan in March and impressed from the off, pressing his case for a permanent switch to the relegated League One outfit. Hutton was always in favour of a switch to the Robins and manager Martin Allen revealed a deal has now been done.
The Cheltenham boss told the club's official website: "David has had 11 years at Tottenham and this is a big move for him. He has the potential to be a top player and he is a very good acquisition for the club."
Allen also revealed Cheltenham will not be offering striker Damian Spencer, who spent the end of the season on loan at Brentford, a new contract at this moment in time, presenting him with the opportunity to look elsewhere.
Former MK Dons boss Allen added: "It is also important for me to sort out our off-field staff and I am hoping to have this all in place by the end of the week along with, hopefully, another signing."