FC United, the only side to have guaranteed their place in the second round, will travel to the winner of BetVictor Northern Premier Division rivals Atherton Collieries' tie with Barrow, which takes place on Tuesday night.
Matlock Town, who surrendered a two goal lead to draw 2-2 with National League side Chorley on Saturday, face a trip to Eastleigh if they can navigate their replay.
Waiting for South Shields if they can get past Southport in their replay is AFC Fylde.
The draw was made live on talkSPORT2 by our National Game Competitions Committee chairman George Dorling and journalist Tony Incenzo and saw 16 ties pulled out of the hat.
Ties will be played on Saturday 11 January 2020 and winning clubs in the second round will pick up £5250 from the prize fund with losing clubs collecting £1750.
The draw
1 Dorking Wanderers v Stockport County
2 Kingstonian v AFC Telford United or Leamington
3 AFC Fylde v South Shields or Southport
4 Royston Town v Hednesford Town or Chester
5 Solihull Moors or Darlington v Harrogate Town
6 Yeovil Town v Tonbridge Angels or Hampton & Richmond Borough
7 Ebbsfleet United v King's Lynn Town
8 Bradford (Park Avenue) or Halesowen Town v Maidenhead United
9 Notts County v Sutton United or Dagenham & Redbridge
10 Chelmsford City v Eastbourne Borough or Salisbury
11 Torquay United v FC Halifax Town
12 Eastleigh v Matlock Town or Chorley
13 Concord Rangers v Bath City or Sholing
14 Farsley Celtic or Altrincham v Barnet
15 Hornchurch v Carshalton Athletic or Aveley
16 Atherton Colleries or Barrow v FC United of Manchester
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