Thompson, 21, was recently released by Sky Bet League Two side Morecambe after signing for the Shrimpers back in the summer.
The goalkeeper made one start for the club, a 2-0 defeat to Fleetwood Town in the Johnstone`s Paint Trophy - having made his league debut a week earlier as a substitute.
Liverpool-born Thompson was previously with Rotherham United and last season had spells on loan at Chelmsford City and Southport in the National League.
However, the Blues have confirmed that keeper Dylan Forth has left the club.
Forth, 18, joined Chester in August after being released by Aston Villa in the summer.