Clarke has broken into the Leeds first team of late under Marcelo Bielsa.
Clarke, 18, has thrived since the appointment of Marcelo Bielsa at Elland Road. The Argentine manager handed the fleet-footed ace his debut as a substitute against Brentford back in October.
He has since clocked up 13 appearances in all competitions for the Whites, starting Leeds' last two matches. He saw out the entire 90 minutes against QPR in the FA Cup, before being taken off after 79 in the win over Derby.
Since breaking into Bielsa's side, Clarke has scored two goals against Nottingham Forest and Aston Villa. He has also created two strikes, with his assist on Friday night setting up Kemar Roofe for Leeds' first in their 2-0 triumph.
Clarke has caught the attention of a handful of Premier League sides, including champions Manchester City. But Austrian boss Ralph Hasenhuttl is now interested in bringing Clarke to the south coast - as per the report.
Southampton are renowned for their faith in youth. On Saturday, Jack Stephens, Yan Valery, Matt Targett and James Ward-Prowse all started the match after spending time in the Saints academy.
Hasenhuttl wants to add Clarke to that system by luring him to the Premier League side this month - but with Southampton fighting relegation and Leeds four points clear, the two sides could swap divisions come May.
Clarke has plenty of time left on his Elland Road contract, after signing a four-year deal in 2017 under Thomas Christiansen to keep him with the Yorkshire club until the summer of 2021.