Black, 52, was filmed apparently naming staff at other clubs who could be induced to pass on information about players to a company for money.
Such payments would be against Football Association rules.
Black, who oversaw Aston Villa’s final seven Premier League matches last season, has denied the allegations.
“It doesn’t take too much to get these people involved,” Black says in the covert recording.
“They won’t have an awful lot of money.”
Former Aberdeen striker Black quotes a figure of “a couple of grand”, which the Telegraph claims was his suggestion of a suitable payment to an unnamed assistant coach at a Championship side in return for introducing players to the undercover reporters’ fictitious agency.
Black denies suggesting that any football official should be paid, saying that his “couple of grand” comment related to a freelance scout that an agency looking to break into the football industry might approach.
Under FA rules, intermediaries “must not give, offer or seek to offer, any consideration of any kind” to a club official “in return for any benefit, service, favour or any kind of preferential treatment”.
A spokesperson for Black said his client did not recall the possibility of bribing officials being mentioned.
“This was not the purpose of the meeting so far as our client understood it. Any suggestion that he was complicit in such discussions is false,” the spokesperson added.
Southampton said they were “fully committed” to investigating the claims and intend to work closely with the Premier League and FA on them.
Black has held permanent managerial posts at Motherwell and Coventry, as well as having caretaker spells at Birmingham, Sunderland, Blackburn, Rotherham and Villa.
He played for Aberdeen and French side Metz and won three caps for Scotland before he became a coach at Celtic.
The allegations about Black are the latest from the Daily Telegraph’s investigation into corruption in football.
On Thursday, Barnsley assistant boss Tommy Wright was sacked over claims he took cash for trying to engineer certain transfers. Wright has “categorically denied” the claims.
Sam Allardyce left his post as England manager on Tuesday, making a “whole-hearted apology” in the wake of allegations in the newspaper that he offered advice on how to “get around” rules on player transfers.
The FA said it has been liaising with The Daily Telegraph and City of London Police over the allegations, adding it would meet police next week.