In the shocking footage the British undefeated middleweight champion also told the woman to slap a passerby and call him a "paedo" if she wanted the drugs.
But while the athlete has apologised for his taunts, the family of the woman, named as 37-year-old Lyndsey Fox, has blasted his behaviour.
Dad Paul Fox told the Mail Online: "My daughter does have drug problems, she has tried to get better in the past but it hasn't worked.
"It is very sad. He is a foolish young man."
Mr Fox said he hadn't heard of Saunders before – but was unimpressed that he had targeted someone so vulnerable.
Speaking from his home in Nottinghamshire, he said: "Police told us she wasn't being hurt in the video but we would still find it too distressing."
Yesterday, the 29-year-old athlete apologised over the video after it surfaced online.
He tweeted: "Apologise to everybody who’s took offence totally in the wrong ‘ can’t do anything But be sorry ‘ banter when wrong ‘ Apologies once again."
In the footage, Saunders pulled over in his £250,000 Rolls-Royce to speak to the woman in Nottinghamshire 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 boxer 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 £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 confirmed they were aware of the video, saying: "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.
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