Regan and Williams have impressed during trials at Meadow Lane this summer and manager Keith Curle has now added them to his squad.
Both players have agreed five-month deals that will expire on January 7 and are eligible to play in the Capital One Cup tie against Bradford on Saturday afternoon.
Regan initially linked up with Notts on loan from Bristol Rovers at the end of the 2010/11 campaign, featuring four times for the club.
The 32-year-old, who has previously worked with Curle at Chester, offers strong competition in both full-back positions.
Williams, also 32, is capable of slotting in at left-back or at the heart of defence and has experienced League One success in the past, having achieved promotion to the Championship in 2009 with Peterborough.
- Keith Curle
- Bristol Rovers