In e-mails and secretly recorded phone conversations, Mr Jones claimed that Sir Alex was so impressed by the BBC's fictional player that he wanted him to come for a two-year trial.
Mr Jones said that the club would coach the player to stardom once a "registration fee" of more than $7,000 (£3,500) was paid.
"As soon as your child comes over here in London, the club will sponsor their food, their accommodation, all the necessary things they need in the club. So after two years of the training, if the player performs well, he will join Manchester United," Mr Jones promised in one call.
In an effort to persuade the undercover reporter that his offer was genuine, Mr Jones e-mailed several fake player registration forms.
The documents were headed with the club's official logo. But one obvious error was that the instructions stated the form should be addressed to legendary United manager, Sir Matt Busby - who died in 1994.
But it appears that Mr Jones was hijacking more than simply Manchester United's reputation.
Paul Jones is almost certainly not the conman's real name.
It appears likely he was hoping potential victims would be fooled into thinking he was a real-life Paul Jones, a London lawyer who has founded a community football project which helps develop poor young players in Ghana.
Many youngsters dream of becoming the next Kanu or Emmanuel Adebayor
In a sting operation the BBC's undercover reporter persuaded the conman to send his assistant, a "Dr Frank Johnson" to meet him at a hotel in the Nigerian city of Lagos to receive the cash in person.
At the meeting Dr Johnson was secretly filmed as he produced another fake registration form.
He assured the undercover reporter that Manchester United were keen to recruit more African players and that his son would be well provided for.
"What we do is give them a good contract. That is why you have to sign as a sponsor," he said.
"We will prepare an apartment for your son and give him special training so he will improve. Then we will set him up with another club or he will play with us."
Dr Johnson's sales pitch was brought to an abrupt end when BBC reporter Gavin Lee and a camera crew approached and confronted him.
Despite the evidence on tape, Dr Johnson initially denied he was a fake. But he later confessed that his real name was John and that he was ashamed of his actions.
Football is seen as a way to escape Nigeria's poverty
He was taken away and questioned by a Nigerian policeman who provided security for the sting.
The BBC team was allowed to examine Dr Johnson's mobile phone and found it contained texts that suggested he was involved with scams involving victims across Africa and Europe. Several people had sent him their bank details.
In the phone's directory the number for the BBC's undercover reporter was listed under "mugu" - the word used by Nigerian fraudsters to describe those that they swindle.
Another conman recorded by the BBC claimed to be Mike Emenalo, the former Nigerian international player who is Chelsea's chief talent scout.
The fake Mr Emenalo was brazen enough to advertise his services on a legitimate sports website.
He claimed to be so impressed by the record of the BBC's fake player that he immediately offered him a trial with the club. This time the fee demanded was $4,000 (£2,000).
Nigerian star Kanu gives his advice to African footballers who want to play in Europe
Nigeria's Sport and Tourism Ambassador John Fashanu commended the BBC investigation for highlighting the perils faced by young African players who want to play abroad.
The former Wimbledon player and England international said there needed to be greater action against the conmen who were ripping off their fellow Africans.
"They have seen an opportunity to make money and they are making money off desperate young Nigerian, Ghanaian, South African, African, Third World country footballers who all want to live of professional footballers like Thierry Henry, Kanu and Jay Jay Okocha," he said.
The English Football Association says it has taken steps to alert the public and other national associations about the scam.
"The FA investigated the complaints and passed on our findings to the relevant law enforcement agencies, whom we continue to liaise with and assist in relation to these matters," it said in a statement.
Often those who fall for scammers like this are from poor backgrounds. Their families scrape together the fees in the vain hope of giving their children a route out of poverty.
Players as young as 12 post their photo, phone number, e-mail address and in some cases even their passport details online in the hope of getting an agent.
Most appear to be unaware that English clubs would never demand money from a player they were considering signing.
These ambitious amateurs seem largely ignorant that under immigration regulations it would be impossible for them to get a British work permit.
They would not qualify as they have not played several games for their own national sides. And it is this naivety which the fraudsters exploit to ruthless effect.
You can hear more about this story in a special documentary on Victoria Derbyshire's show on BBC Five Live at 0900BST on 15 July. BBC World Service listeners can hear it on Assignment on 17 July.
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