The Robins, who will be playing in League Two alongside the Stags next season, dismissed head coach Luke Williams following their relegation earlier this month.
Radford told the official Mansfield website: "Steve is fully committed to Mansfield and we will allow for no distractions as we set about our work to mount a serious promotion challenge in 2017-18.
"I have refused Swindon Town permission to talk to our manager, who is under contract at One Call Stadium, and work continues behind the scenes at our club as we put together a squad which we believe will be highly competitive in Sky Bet League Two next season.
"Steve has my full backing ahead of next season as we aim to fulfil the dreams of a town and its people. There is a tremendous buzz around One Call Stadium right now as we build for next season."
Evans replaced Adam Murray at the One Call Stadium in November 2016.