The 19-year-old winger is on loan at Whaddon Road from Tottenham until the end of the season, when he is due to be released at White Hart Lane.
Hutton's impressive displays have already led to manager Martin Allen singing his praises and hinting "he could be one we could take next year".
The Republic of Ireland youth international is desperate to pen a deal with the all-but relegated Robins, and he is optimistic his performances are giving him a good chance of being at the club come August.
"There are only a couple of weeks of the season left and not many games, so I am doing it for me and I have to impress Cheltenham and as many people as I can," he told the Gloucestershire Echo. "I will keep doing my best and I definitely think I am going the right way about getting a contract here.
"I am really enjoying my football again, the lads have been great and I am grateful to the manager for giving me this chance."