An eyewitness said they saw him in the early hours of Tuesday in the Hyatt Regency hotel bar in Birmingham with the man during the party's conference.
Mr Burns has denied any wrongdoing.
The BBC has not spoken to, or heard the account, of the individual the ex-minister was seen with.
Downing Street has not commented further on the circumstances surrounding the sacking.
The eyewitness claimed that the minister had his hand on the young man's thigh, and that there were several onlookers.
They added that Mr Burns was told at the time by an onlooker to stop what he was doing, although this is disputed by friends of the Bournemouth West MP.
A second source, with knowledge of the disciplinary process, corroborated this account and confirmed this incident was what led to him being sacked as well as losing his whip.
The BBC understands that a third party — not the man seen with Mr Burns — raised this incident with party whips, who look after discipline among MPs, and that this complaint led to the suspension of his whip and his sacking as international trade minister.
Somebody with knowledge of the disciplinary process told the BBC that concerns were also raised about "inappropriate comments" made by Mr Burns on that night.
Allies of the MP claim he had injured his ribs the weekend before the party conference and was on heavy medication to manage the pain.
They suggested the prescribed medicine made the effect of the alcohol worse.
It is not disputed by the former minister's friends that he had been drinking or that he flirted with the young man who had joined him, who, we understand, was not known to Mr Burns.
The MP was sufficiently drunk that he had to later be taken back to his hotel by a friend.
We understand that Mr Burns is strongly of the view that the flirting was consensual, but the BBC has not yet spoken to the man the former minister was with to hear his account of events.
An ally of the sacked minister told the BBC: "He feels imprisoned without any of the process of the police or a trial."
The ally added that his career had been ruined, as even if he were to be exonerated and the whip was restored, he would have lost his ministerial role.
The prime minister has not spoken to Mr Burns since his sacking.
Another suggested what had happened suggested a "morality police" could now act in judgement on the behaviour of others, irrespective of what those involved made of it.
One source suggested that after a "diabolical conference" and an "atrocious start" to Liz Truss's time as prime minister, she was determined to use this incident as an example of her demonstrating her authority over her party and government.
The source suggested that given former prime minister Boris Johnson had "vacillated" over the case of Christopher Pincher, Ms Truss was determined to be seen as decisive, irrespective of due process.
A Conservative Party source said: "The prime minister has been clear that the highest standards in public life must be maintained."