The 19-year-old, who was linked with a move to Liverpool, has been signed for a deal worth 55 million pounds ($69.96 million), according to British media.
The BBC reported that Chelsea agreed a fee of 53 million pounds ($67.32 million) for the 19-year-old.
The 28-year-old made over 400 appearances for Southampton and captained the club last season when they were relegated from the Premier League after finishing bottom.
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.
Jacob Ramsey netted his first goal of the season to give Aston Villa a scrappy 1-0 home victory over Southampton in the Premier League on Friday and snap a run of four games without a win.
Brentford's match against Manchester United on Tuesday was postponed after United's training ground was closed due to an outbreak of positive cases.
Southampton dumped FA Cup holders Arsenal out of the competition on Saturday, winning 1-0 courtesy of a Gabriel own goal.
Southampton moved into first place in England’s top flight for the first time since 1988 after a comfortable 2-0 home victory over Newcastle United.
Ralph Hasenhuettl sees this as an opportunity to hand deadline day signing Theo Walcott his debut at Stamford Bridge.
Son’s four goals were set up by the team striker, Harry Kane, who added the fifth goal for Spurs minutes before the final whistle.
Palace will begin their Premier League campaign against Southampton with eight first-team players unavailable.
Tottenham Hotspur have agreed a deal in principle with Southampton to sign Denmark midfielder Pierre-Emile Hojbjerg with full back Kyle Walker-Peters going the other way, British media reported on Tuesday.
Manchester United’s Champions League qualification hopes suffered a blow after an equaliser deep in stoppage time from Michael Obafemi earned Southampton a 2-2 draw in the Premier League at Old Trafford on Monday.
Southampton's win over fellow strugglers Fulham on Wednesday put the spring back in their step but manager Ralph Hasenhuttl says his team's fight for Premier League survival is far from over.
Southampton manager Ralph Hasenhuttl is not afraid to go toe-to-toe with the Premier League's top sides as he looks to lead the club out of the relegation zone, the Austrian said.
It may be a clash at the wrong end of the table, but Southampton's visit to Burnley on Saturday pits two resurgent teams against one another, with neither set to take a backwards step.
Pochettino hopes to emulate the longevity of Wenger, who was in charge at Arsenal for 22 years before leaving at the end of last season.
Southampton striker Charlie Austin has been charged after appearing to make an obscene gesture towards Manchester City supporters, the Football Association (FA) said on Thursday.
Southampton striker Charlie Austin has been charged after appearing to make an obscene gesture towards Manchester City supporters.
Southampton sacked Mark Hughes as manager on Monday with the Premier League club mired in the relegation zone after securing only one victory this season.