The 26-year-old, who joined the League One outfit on a temporary basis from Cardiff City earlier this year, is due to return to his parent club at the end of the season.
However, having previously worked with manager Nigel Clough at Burton Albion and Derby County, the full-back has expressed his desire to remain at Bramall Lane.
"I only really came here for the gaffer, having played for him at my two previous clubs, but I'm enjoying every minute of being here. I feel privileged to have been given the chance to play for this club," Sky Sports News quotes him as saying.
"The lads in the dressing room are fantastic and the fans are different class. I would really welcome the chance to be here again next season because I have fallen in love with this club."
Brayford has made 16 appearances for the Blades.