Holders Manchester City will travel to Sheffield Wednesday.
All eight ties will be played across three midweek nights from 3-5 March.
Rooney, who is United and England’s all-time record goalscorer, joined Championship club Derby County as a player-coach in January from MLS side DC United.
Now 34, he played 559 times for United, scoring 253 goals.
Since leaving Old Trafford he has faced his former employers twice, losing both games during the 2017-18 season, while in a second spell at Everton.
Elsewhere in the draw, League One Oxford United face a trip to Championship leaders West Brom if they can follow up their shock draw at Newcastle United with victory in the replay, while Reading or Cardiff will host Sheffield United.
Leicester will host Coventry or Birmingham in an all-Midlands affair and Southampton or Tottenham will play Norwich.
Why are fifth-round ties being played in midweek?
In a change to the usual format, all eight ties will be played across three midweek nights from 3-5 March, with no replays.
If any of the ties are level after 90 minutes there will be 30 minutes of extra time and, if necessary, penalties to determine a winner on the night.
It is the second season running that fifth-round replays have been removed from the competition in an attempt to alleviate fixture congestion.
In addition, the Premier League is having a mid-season break in February, with each team getting a weekend off.
Premier League winter break: All you need to know
Four Premier League matches will take place on the opening weekend of the inaugural winter break, with six more the following weekend.
All 10 matches have different kick-off times, meaning none of the games will be played at the same time.
The schedule was agreed in June 2018 between the Football Association, the Premier League and EFL.
FA CUP THIRD ROUND MATCHES
Sheffield Wednesday v Manchester City – Buy FA Cup Tickets
Reading or Cardiff City v Sheffield United – Buy FA Cup Tickets
Chelsea v Shrewsbury Town or Liverpool – Buy FA Cup Tickets
West Bromwich Albion v Newcastle United or Oxford United – Buy FA Cup Tickets
Leicester City v Coventry City or Birmingham City – Buy FA Cup Tickets
Northampton Town or Derby County v Manchester United – Buy FA Cup Tickets
Southampton or Tottenham Hotspur v Norwich City – Buy FA Cup Tickets
Portsmouth v Bournemouth or Arsenal – Buy FA Cup Tickets
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