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West Ham will host Manchester City in the pick of the all-Premier League ties, with Everton set to welcome Leicester City.
Holders Manchester United will entertain Reading, managed by former United defender Jaap Stam.
National League Sutton United will face neighbours AFC Wimbledon.
The 32 third-round ties will be played from 6-9 January 2017.
Newport are a fan-owned club and chairman Gavin Foxall said of the prospect of a tie at Anfield: "It would absolutely be a massive financial boost for us.
"Certainly for a club like ours it would secure the future for a season or two at least.
"A game at Anfield against Liverpool would rank right up there in the club's history."
While Liverpool's opponents are yet to be decided, the biggest beneficiaries of the definite ties look to be Championship Preston, who welcome 2014 and 2015 winners Arsenal.
"It is great for the football club," manager Simon Grayson told the North End club website.
"Being at home, it gives your own fans the opportunity to see some world-class players and they don't come much bigger and better than this Arsenal squad at this time."
For holders Manchester United, the visit of Reading will see a return to Old Trafford for former defender Stam, who was part of the 1999 treble-winning team.
"I look forward to that," said Stam's former United team-mate Andy Cole on BBC Two. "He's done very well at Reading. I'm really pleased for him."
The lowest ranked club left in the competition, Stourbridge United, face a potential trip to League Two Wycombe Wanderers, should the seventh-tier Northern Premier League Premier Division club beat Northampton Town in a rearranged fixture.
Of the other non-league sides through to the third round, Barrow will host Rochdale while either Halifax or Eastleigh go to Brentford.
"I think the chairman will be really happy, because at least we're at home," said Barrow manager Paul Cox.
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League Two Newport County or Plymouth Argyle will travel to Anfield to face Premier League Liverpool in the FA Cup third round.