The visitors will enter the match off the back of a 3-0 win over Birmingham City in the Championship, while Peterborough beat Derby County 3-2 in their last fixture in League One.
Match preview
Peterborough's main focus for this season is to secure a return to the Championship; they came close last term, finishing sixth in the division, but they were incredibly beaten by Sheffield Wednesday in the semi-finals of the playoffs despite winning the first leg of their last-four encounter 4-0 at home.
The Posh have since moved on from that disappointment, and they are currently up in third spot in the League One table, boasting 49 points from their opening 25 matches of the campaign.
Darren Ferguson's side, who recorded a 3-2 win over Derby in the league on Monday, are just two points behind second-placed Bolton Wanderers and only four from leaders Portsmouth.
Peterborough will bring a 10-game unbeaten run into this match, while they have not lost in a competition other than the EFL Trophy since the start of November.
Ferguson's team beat Salford City on penalties following a 4-4 draw in the first round of this season's FA Cup before recording a 2-1 victory over Doncaster Rovers in the second round.
Leeds, meanwhile, also have their eyes on promotion this season, with the Whites bidding to return to the Premier League following their relegation in 19th position last term.
Daniel Farke's side are fourth in the Championship table, picking up 48 points from their opening 26 matches of the campaign, which has left them nine points ahead of seventh-placed Hull City, but they are seven points behind second-placed Ipswich Town.
Leeds suffer back-to-back losses to Preston North End and West Bromwich Albion on December 26 and December 29 respectively, but they recorded a 3-0 victory over Birmingham last time out.
The Whites progressed to the fifth round of last season's FA Cup, where they were beaten 2-0 by Fulham, but they have history when it comes to winning this competition.
Indeed, Leeds lifted the FA Cup at the end of the 1971-72 campaign courtesy of a 1-0 victory over Arsenal in the final, with Allan Clarke netting the only goal of the contest.
Peterborough United FA Cup form:
D
W
W
Peterborough United form (all competitions):
W
W
W
D
D
W
Leeds United form (all competitions):
L
D
W
L
L
W
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Team News
Peterborough head coach Ferguson is expected to name a strong side for this contest, with the team potentially unchanged from the one that started on New Year's Day.
Ephron Mason-Clark is the team's leading goalscorer this season with 11, and he is set to again feature in the final third, while Kwame Poku, who has netted nine times this term, should also start.
Ricky-Jade Jones has seven goals this season and is likely to lead the line, while there should be another start in midfield for Archie Collins.
As for Leeds, Luke Ayling, Pascal Struijk, Sam Byram, Jamie Shackleton and Stuart Dallas will miss the match through injury.
The Whites will also be unable to call upon goalkeeper Illan Meslier, who is suspended due to the red card that he received in the clash with Preston on Boxing Day.
Head coach Farke will make changes to the side that started against Birmingham, but there should be another start for Crysencio Summerville, who is the team's leading goalscorer this season with 12.
Junior Firpo, Joel Piroe and Wilfried Gnonto are likely to be among the players introduced into the away side's starting side for this FA Cup third-round contest.
Peterborough United possible starting lineup:
Talley; Kioso, Knight, Edwards, Burrows; Kyprianou, Collins; Poku, Randall, Mason-Clark; Jones
Leeds United possible starting lineup:
Klaesson; Gray, Rodon, Cooper, Firpo; Ampadu, Kamara; Summerville, Piroe, Gnonto; Bamford
We say: Peterborough United 2-3 Leeds United
A bit of a bold prediction in terms of the scoreline, but we just fancy goals in this contest, with Leeds just coming out on top. Peterborough have so much quality in the advanced areas, and the same can also be said for Leeds, even if Farke, as is expected, rests some important players.