Pochettino is thought to be in talks with Tottenham Hotspur, who have offered the Argentine a three-year deal to take over at White Hart Lane, according to a report.
The Saints are said to have been lining up possible replacements for Pochettino for some time.
McClaren has enjoyed a successful season with Derby, having led the club to the Championship playoff final, where they lost against Queens Park Rangers.
The former England manager has the backing of Southampton executive director Les Reed, according to the Daily Star.
McClaren took charge of the Rams in September 2013.