Andy Hughes, a die-hard Manchester United fan from Colwyn Bay, north Wales, filmed Carragher spitting at his daughter after his side beat Liverpool 2-1 at Old Trafford on Saturday.
Police are understood to have spoken to Mr Hughes about filming on his phone while behind the wheel today.
He has been slammed by road safety campaigners for filming the incident on his phone while driving.
Veteran Radio 2 presenter Jeremy Vine – a keen cyclist and road safety campaigner whose evidence
secured the conviction of a road rage driver last year – is among those to have weighed in on the issue, arguing the motorist should be sent to prison.
He wrote on Twitter: "Spitting: disgusting, possibly criminal, certainly sackable.
"Filming on a mobile while driving a car with a child in the back, on a motorway, paying no attention whatsoever to the road: straight to jail."
It is understood Greater Manchester Police have spoken to Mr Hughes but do not plan to prosecute him. Using a phone at the wheel is illegal and drivers can be given a £200 fine and six penalty points.
The former England defender was forced to watch the shocking footage live on Sky News and issued a grovelling public apology yesterday as he fights for his job.
He said: "I have no excuse. My biggest regret is for the 14-year-old girl to be caught in the middle of this, my altercation with the father."
Asked if he deserves to keep his job, he replied: "There's no doubt what I have done is disgusting, I'm getting vilified and rightly so. I would hope five seconds of madness will not take over everything of what I've done – I don't feel like that's a real representation of me."
Mr Hughes had previously denied abusing Carragher but speaking after the 40-year-old was suspended, said he does not want him to lose his job.
He told the Mirror: "We wanted an apology and an explanation. He seems contrite. Everyone makes mistakes, we are all human."
The incident took place near Manchester United's Old Trafford ground after Carragher watched his former side Liverpool lose 2-1 on Saturday.
The AA has also said Mr Hughes should be sent on a drivers awareness course and had put other road users at risk. President Edmund King said of the incident: "It is a serious offence to use a hand-held phone at the wheel which one driver certainly seems to be doing. The tactical display from both drivers, on attack and in defence, is diabolical."
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