O’Brien played 16 times for Swansea after joining on loan in the mid-season window, and played a key role as they pulled clear of relegation danger to finish 11th.
He also had loan spells at Los Angeles FC, Middlesbrough and DC United after falling out of favour at Forest.
O’Brien’s signing would be another statement of intent by Wrexham, who are owned by Hollywood stars Ryan Reynolds and Rob McElhenney, as they prepare for life in the Championship after three successive promotions.
The north Wales club have already brought in goalkeeper Danny Ward from Leicester City, forward Ryan Hardie from Plymouth Argyle, left-back Liberato Cacace for a club-record fee from Empoli, midfielder George Thomason from Bolton Wanderers and forward Josh Windass, who was a free agent after leaving financially troubled Sheffield Wednesday.
