The two teams locked horns at Rodney Parade on December 2 but could not be separated, with a 1-1 draw meaning that a replay is required to determine which side will progress to round three.
Barnet were on the verge of being eliminated in the FA Cup earlier this month, with Shane McLoughlin sending Newport County into the lead shortly before the break, and his goal separated the two teams until the 89th minute, when Daniele Collinge netted a leveller for the visitors.
The Bees have already played four times in this season's FA Cup, beating Aveley after a replay before recording a 1-0 victory over Curzon Ashton in the first round of the competition at the start of November.
Eastleigh are the team waiting in the third round of the tournament, so a victory for Barnet in this match would place them in a strong position to reach round four, but Newport County will also be thinking the same.
The home side will enter this contest off the back of a 4-0 win over Maidstone United in the FA Trophy on Saturday, but they are on a poor run of form in the National League, losing each of their last four.
Barnet remain in a healthy position in the National League table despite their recent defeats, sitting third, but they are now 11 points behind the leaders Chesterfield.
Newport County, meanwhile, will enter this match off the back of a 2-1 defeat at Tranmere Rovers on Saturday, with the result leaving them in 15th spot in the League Two table on 25 points.
The Port have won two of their last four games in all competitions, while they recorded a 2-0 victory over Oldham Athletic to progress to this round of the FA Cup.
Graham Coughlan's side will feel that they should have got the job done against Barnet in their first meeting, especially as this match comes ahead of an important game in League Two against Grimsby Town on Saturday.
Newport County have twice managed to reach the fifth round of the FA Cup, including in 2018-19, when they were beaten by the eventual winners Manchester City.
The South Wales outfit have not actually beaten Barnet since December 2015, though, which remains their only victory over the Bees in nine previous meetings.
Barnet head coach Brennan could name the same side that took to the field for the first whistle of the contest with Newport County last weekend.
Nicke Kabamba has been in impressive scoring form this season, finding the back of the net on 14 occasions, and the 30-year-old should continue in the final third of the field.
Collinge has seven goals from central defence, and he should also continue in the XI, while the experienced Gary Hooper is also likely to be named in the side.
As for Newport County, Will Evans has scored 14 times during an excellent campaign, and the 26-year-old is set to again feature in an attacking area for the League Two side.
Head coach Coughlan is expected to name a strong side for this game, especially as Eastleigh are waiting in the next round, with the majority of the away team's big-hitters set to feature.
Barnet possible starting lineup:
Walker; Oluwo, Collinge, Okimo; Cropper, Hartigan, Brunt, Pritchard; Hooper, Kabamba, Kanu
Newport County possible starting lineup:
Townsend; Baker, Delaney, Clarke; Payne, Morris, Bennett, S McLoughlin; Evans, Bogle, Palmer-Houlden
We say: Barnet 1-2 Newport County
The first meeting between the two sides last weekend was a tight affair, and we are expecting this match to follow suit. A case can be made for both to win it, but we just have a feeling that Newport County will be able to edge a close contest in order to advance to the third round of the competition.
For data analysis of the most likely results, scorelines and more for this match please click here.