"I'm delighted to have signed Sam," Saunders told Wrexham's website.
"He has had a very good grounding in the game with City and typifies the kind of player I feel can help take the club forward.
"He is young, ambitious and also keen to continue his education as a footballer."
Williamson added: "It's a wrench leaving City after so long but I've enjoyed my time on loan Wrexham and that has made it much easier to make the switch.
"I don't see this as a step down, rather a chance to further my career."
Saunders has also offered contract extensions to Wales Under-21 striker Marc Williams and Andy Fleming.
Rookie striker Williams has been instrumental in Wrexham's revival under Saunders where the North Wales side have lost just once in 12 league games.
The 20-year-old's 11 goals this season have helped the Racecourse club to the verge of the Blue Square Premier play-offs and league clubs have taken note.
"The phones have been ringing for our players," Saunders told the Wrexham Evening Leader.
"I have to think about that but I want to keep players who I think have a future here."
Saunders will meet with Williams' agent this week while midfielder Fleming, whose current deal expires in the summer, has been offered a 12 month extension.
"I have made informal offers to Andy and Marc," confirmed Saunders. "Hopefully we can tie those two up."
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