The United manager faces competition from Manchester City, and is believed to have stepped up his pursuit of the 18-year-old centre-back to try and beat the Premier League champions to the signing of one of the most highly-rated youngsters in French football.
Zouma was promoted to the St Etienne first-team last season while still just 17-years-old, and the young defender has already made 27 league appearances for the Ligue 1 club.
The 6ft 2in centre-half is a France under-19 international, and the report claims St Etienne value him at £9m.
Manchester United have a strong track record when it comes to spending money on highly-rated young stars - they paid close to £20m for David De Gea last summer, and paid £25.6m for a 18-year-old Wayne Rooney in 2004, while an 18-year-old Cristiano Ronaldo cost around £12m in 2003.
But Manchester City are expected to rival United for the signature of Zouma. Roberto Mancini paid close to £13m for 19-year-old Matija Nastasic earlier this summer, but the report claims he's interested in Zouma as well.
The Etihad Stadium club announced a £200m plan to revamp their training and academy facilities this season, and are expected to invest heavily in youth over the coming years.