The Hawks have been drawn to face Hampshire rivals Basingstoke in the FA Trophy.
Both sides are currently struggling in the National League south, with Basingstoke currently bottom of the table and the Hawks sitting in 16th.
But Lee Bradbury’s side will be confident of progressing beyond the third qualifying round, having been handed home advantage and fresh from two successive 3-1 league wins.
Finallists in 2014, Gosport Borough have been handed an away draw against Southern League premier division Cirencester Town.
Meanwhile, Ryman League Bognor have been drawn at home to Bath City.
All ties will be played on Saturday, November 28.