Forest will host the first leg after beating Wolverhampton Wanderers in a penalty shootout while Marcus Rashford's late double led Manchester United to a 3-0 victory over third-tier Charlton Athletic on Tuesday to reach the last four.
Southampton, bottom of the Premier League, pulled off a shock 2-0 win over champions Manchester City and will host Newcastle, who beat Leicester 2-0, in the first leg of their tie.
The semi-finals will take place in the weeks commencing January 23 and January 30.