Holders Arsenal will travel to Championship (second tier) Brighton & Hove Albion while joint Premier League leaders Chelsea will face Championship Millwall or third tier Bradford City on Jan. 24 or 25.
Champions Manchester City are at home to Championship side Middlesbrough and Birmingham City entertain West Bromwich Albion in a Midlands derby.
"It's a game to look forward to," Cambridge United head coach Richard Money, whose side returned to the league only in 2014, told the BBC of a match against a side 79 places above them. "It is reward for everyone's hard work at the club.
"Everyone's suffered during nine years out of the Football League but a lot of people have worked hard and now we've got a draw like this. We're delighted."
The match will be Manchester United's second in a row against lowly opponents after Louis van Gaal's team beat famed League One giant-killers Yeovil Town 2-0 in the third round on Sunday.
If Tottenham Hotspur win a replay against Burnley after Monday's third round match ended 1-1, they will play Leicester City in another all Premier League clash.
Spurs defeated Burnley in the 1962 FA Cup final and overcame Leicester to win the trophy in 1961.
Premier League Aston Villa host Championship leaders Bournemouth while Liverpool, who beat fourth tier AFC Wimbledon 2-1 on Monday, will entertain Bolton Wanderers.
That match could see Bolton's former Liverpool striker Emile Heskey make a return to Anfield.
(Reporting by Ed Osmond, editing by Pritha Sarkar/Alan Baldwin)
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