The 27-year-old had been offered a one-year deal at Underhill but is expected to put pen to paper on a contract to join the Magpies this week.
Bishop becomes the second midfielder in as many days to agree a move to Meadow Lane after Ricky Ravenhill on Tuesday, and manager Ian McParland believes the two players will compliment each other well.
"It has been an excellent couple of days and the message is that despite all the disappointments of last season, good players still want to come and join us," McParland told the club's official website. "I couldn't be more pleased. One of the priorities this summer was to get more 'leaders' on board and we have already added two more in Ravenhill and Bishop.
"(Assistant manager) Dave Kevan and I trailed Neal for much of last season and we think he has a real presence about him. We are sure he can make a big contribution and believe he will blend well with Ricky Ravenhill.
"He's a six-footer who wins headers in both boxes, is a good tackler and makes 'easy' passes. If we need him to go forward he can and he is just as adept in the holding role."