Hastings, who made 64 appearances for Scotland between 1986 and 1997, wrote on Twitter: “I am really upset about this. These despicable idiots are a disgrace.
“We need to find them, name & shame them, & fine them for public disorder & harassment. Disgusting behaviour from a bunch of drunk idiots.”
Jones admitted on Wednesday that he had “feared for his safety” while travelling by train from Edinburgh to Manchester on the morning after England’s Calcutta Cup loss to Scotland at BT Murrayfield.
The Australian was travelling on his own in standard class and had been invited to Old Trafford as a guest of Sir Alex Ferguson at Manchester United’s match with Chelsea on Sunday afternoon.
Video footage taken by the BBC showed Jones posing for photos with supporters at Manchester Oxford Road station while en route to the stadium, before the group turned on the 58-year-old and launched a tirade of abuse at him.
In the clip, one individual shouts at Jones: “What happened last night, ya baldy c***?”
A station worker can be seen attempting to usher the four men away from Jones as he gets into a waiting car, before the fans approach the vehicle and open the rear passenger side door, chanting: “Scotland, Scotland, Scotland.”
One of the men can then be heard calling Jones a “f*****g English b*****d”.
British Transport Police confirmed they were investigating a separate incident involving Jones, understood to have taken place on a train between Manchester and London.
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Officers were set to meet the train as it arrived at Euston Station on the Sunday evening, amid reports that the England head coach had been subjected to verbal abuse from football fans while on the train.
Scottish Rugby branded the fans’ behaviour “disgusting,” and in a strongly-worded statement said: “Scottish Rugby is appalled by the verbal abuse suffered by Eddie Jones. The disgusting behaviour of those involved does not represent the values of our sport or its fans.
“The dignity Eddie and the England team showed on Saturday is in stark contrast to this ugly incident.”
Jones said earlier this week that he wouldn’t risk travelling on public transport in the future after the two incidents.
He said: “I’m a human being. I don’t consider myself any different from anyone else, so for me to travel on public transport I thought was okay. But I’ll make sure I won’t in future. It’s as simple as that.
“I can’t because it was shown on Sunday what happens when I do. That’s the world we live in. I was massively surprised. It wasn’t comfortable.”
Jones also voiced his concern at comments made by Gavin Hastings and current cap Simon Berghan in the lead-up to the Calcutta Cup clash last weekend.
Berghan claimed that “everyone hates England” while Hastings, capped 67 times for Scotland, said in an interview: “As a supporter of one of his opponents, you just want to rub his face in the dirt.”
Jones believes that Hastings and Berghan should have chosen their words more carefully, adding: “It magnifies that if you’re in a position of responsibility you’ve got to be careful what you say.”
Insisting that he always tried to “do the right thing by the fans” Jones said: “I never knock back a request for a selfie unless I’m racing to somewhere, but if this happens then you’ve got to have a look at your own safety.”
When asked if the abuse he was subjected to was verbal or physical, Jones stated that it had been “a bit of both.”
He continued: “It’s part of the challenge. As an Australian coaching England, there were always going to be challenges and that’s just one of them.”
Former Scotland international John Beattie also condemned the fans’ behaviour, adding: “I am so angry. Eddie Jones, you shouldn’t have to go through this. So sorry.”
Beattie later added: “I feel ashamed of this. All right-minded Scots will be ashamed of the abuse of Eddie Jones by drunken Scots.
“Most of us regard our country as having moved on from a myopic view of England.”
Former Wales fly-half Jonathan Davies appeared to suggest that he too had been subjected to abuse, adding: “We all get it. Did the same journey Saturday night last year. Always brave in gangs as well.”
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