The two Premier League giants must negotiate tricky fifth-round ties before meeting at Old Trafford, while Manchester City host Barnsley as reward for beating Leeds United 4-0
The FA Cup holders convincingly beat Brentford 4-0 at Stamford Bridge on Sunday afternoon and face Middlesbrough at the Riverside Stadium on February 27, while United host Reading on Monday night at Old Trafford.
Oldham, the lowest-ranked side left in the cup, have to find a way past Everton in their fifth-round replay at Goodison Park after Matt Smith salvaged the tie at Boundary Park on Saturday evening, but will be rewarded with a home tie against Wigan should they do so.
The Premier League champions, Manchester City, face Barnsley at the Etihad Stadium, while Millwall entertain Blackburn, who sensationally eliminated Arsenal on Saturday, at The Den.
All ties will be played on the weekend of March 9 and 10.
Sixth-round draw in full:
Oldham or Everton v Wigan
Manchester City v Barnsley
Manchester United or Reading v Middlesbrough or Chelsea
Millwall v Blackburn