The 22-year-old left-sided player, who can also operate on the wing, made 30 appearances for David Howell's Ryman Premier League team last season, scoring four goals.
He has already trained twice with the Cards and manager Graham Baker will continue to monitor his progress for around two weeks before deciding whether to offer him a deal.
“In my eyes Ryan is a professional player who should be playing league football,” said Borough boss Howell. “Injury interrupted his season least time and he never contributed as much as he would have liked.
“He can play at a higher level and all the attributes are in place for him to achieve that. If anyone is looking for a left-sided attacking defender then Ryan is your man.”
Watts came through the youth ranks at Brentford before joining Carshalton Athletic and then Borough two years ago.
He was voted the club's player's player and supporters player of the season in the 2008/09 campaign.
Meanwhile, Howell will cast his eye over hopefuls at an open trial tonight (Thursday). Defenders would appear to be top of his shopping list following the departure of Tobi Jinadu to Kingstonian and Woking's interest in Watts.
One player Borough will be without until next month is goalkeeper Andre Baptiste. The tall shot stopper has flown out to his country Grenada, who he has represented, to help with goalkeeper coaching ahead of a tournament in October.