The 18-year-old, whose twin brother Josh also plays for the Canaries, made his senior debut for the Premier League club last month against Fulham, having been a key part of Norwich's FA Youth Cup-winning side last season.
Murphy, an England Under-19 international, has impressed for the Canaries' Under-21s this season.
"I think that's one we've almost done," Robins manager Mark Cooper told BBC Sport. "We can't sign anyone until Saturday, but I think it's more or less agreed."
Swindon Town are currently in eighth place in this season's League One table.