Notts new manager Craig Short, who has only been in charge at Meadow Lane for two days, said: "I am delighted to have secured my first signing.
"I am looking to put together a squad that can challenge in League One."
Burch turned professional with Tottenham Hotspur in July 2002 and spent a further five years at the club.
He had loan spells with Woking, Stevenage, West Ham, Bristol City and Barnet, before signing for Sheffield Wednesday in 2007.
The former England Under-20 goalkeeper was released by the Owls after just one season and three competitive matches, and signed for Lincoln in July 2008.
Source: BBC Sport
Source: BBC Sport