Tottenham are reportedly confident of tying up deals for the French midfielder and last season’s top scorer.
Discussions are believed to have taken place between the clubs.
Saints are understood to want well in excess of £20m for Schneiderlin while Rodriguez, despite being sidelined for the start of the season with an ACL injury, will also command a handsome fee.
Schneiderlin joined the club from Strasbourg in 2008 and played a key role in their rise from League One to the top-flight.
The 24-year-old represented France at this summer's World Cup and has attracted interest from Arsenal, although he is understood to favour a move to Spurs so he can work with Pochettino again.
The former Argentina international is also in for Rodriguez.
The 24-year-old has been at St Mary's since joining from hometown club Burnley two years ago and made his Three Lions debut in November's friendly against Chile.
Their exits would be another hammer blow to recently-appointed manager Ronald Koeman, who is also expected to sell disruptive club-record signing Dani Osvaldo in the coming weeks.
Work is afoot to bring in replacements but so far only two have arrived, with Dusan Tadic joining from FC Twente before Graziano Pelle arrived from Koeman's former club Feyenoord.