The British undefeated middleweight champ also told her to slap a stranger and call him a "paedo" if she wanted the drugs.
Saunders pulled over in his £250,000 Rolls-Royce to speak to the woman before goading her.
A source told The Sun Online: “He pulled over deliberately to taunt this woman after spotting her and thinking it would be a laugh."
After engaging the woman in conversation, the undefeated middleweight champion, 29, said: “Listen, let me tell you one thing: you are not going to tell anyone if I give you this crack.
“Swear on your granddad's grave.”
The woman, who appears to be suffering from an addiction issue, complies and pleads for the alleged drugs to be handed over.
In the video, he adds: “Tell me the truth. How much do you need it.”
She responds: “Quite a lot.”
He then tells her: “If you want it, I’ll give it to you. I’ll give you a £150 worth.”
The sickening video is believed to be filmed from inside the boxer's Rolls-Royce.
He then asks the lady to perform a sex act on his front seat passenger and after haggling he asks her instead to punch an innocent man walking along the street and call him a paedophile.
He says: "I know him, he's a paedophile, punch him in the face and I'm going to give you £150-worth.
"Punch him in the face, go on I'm going to give it to you right now, punch him he's a paedophile, punch him."
She then slaps the innocent bystander and shouts “paedophile” at him.
Saunders then tells her he is not a sex-offender and pulls off laughing as she apologises to the man.
Nottinghamshire Police said they were aware of the "sickening" clip and were now investigating.
A force spokesman said: "[We are] aware of a video circulating, where a woman is offered drugs in exchange for committing an act of violence, and potentially committing a sex act in public.
"Officers will be looking to identify and speak to all of those who feature in this sickening video, and a full investigation will be carried out into the number of offences that have been captured on film, including the offer to supply Class A drugs, aiding and abetting an assault, assault and public order offences.
"Incidents of this nature will not be tolerated in Nottinghamshire, and we would urge anyone who witnessed this incident, or who has any further information about those responsible to come forward."
There is no suggestion in the video that Saunders had any drugs, or any genuine intention of supplying them to the woman.
It is the latest controversy for the boxer – who will defend his WBO middlewight crown next month against Demetrius Andrade.
He was filmed being thrown out of a Nando's by members of boxer Deontay Wilder's entourage last month before Tyson Fury's fight in Belfast for allegedly throwing chicken at Wilder.]
Last year he also mocked a shop worker online for having long hair and wearing hoop earrings. He was branded a bully after uploading the picture to his social media account.
After his second round loss in the 2008 Olympics he was suspended after confronting a French woman in a sexually explicit and abusive manner while staying in a hotel with the British team.
Responding to the footage, the General Secretary of the British Boxing Board of Control, Robert Smith, said: "I have just seen the video and I can tell you we find it disgusting."
He added: "I am speaking to members of the board now…I am keen to get on with this as soon as possible."
Saunders' spokesman did not respond to The Sun's request for comment.
But a member of his team claimed the video was “a fake-news prank” designed to “drum up publicity” for his forthcoming fight.
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