The clip, tweeted on Sunday morning, reportedly shows Liverpool supporters taunting the Toffees defender, re-working a song sung by Everton fans in honour of the 28-year-old.
Republic of Ireland international Coleman may not play again in 2017 after breaking his leg in two places during his country’s World Cup qualifier against Wales last weekend.
Evertonians, and wider football fans, were quick to send their best wishes to the 28-year-old, who faces a lengthy rehabilitation stint.
Jurgen Klopp also used his programme notes before Saturday’s 3-1 Reds win over their rivals to wish Coleman the best.
But fans in the video sing “How’s your leg Seamus Coleman? I hope you never play again’.
Les Lawson, the chairman of the Merseyside branch of the Official Liverpool Supporters Club, said it was not representative of the wider Liverpool fan-base.
He said: “Straight away after the injury everyone was hoping he makes a speedy recovery. It was a horrific tackle.
“It can easily be one of your own players and you should never mock a player who gets hurt like that.
“The majority of Liverpool fans wish him well and hope he is back to playing again soon.
“I remember, despite what was claimed, that when Alan Smith broke his leg and was stretchered off at Anfield the whole ground stood and applauded him.”
Everton fans on social media reacted with anger at the video.
In his notes Klopp wrote: ‘I would like to pass on my regards and best wishes to Seamus Coleman, who due to very unfortunate circumstances is not available to play today.
“I do not know him personally, but as a football supporter I love his character, his positive approach to the game and his commitment to his club. There is a lot to admire about this player and it is his personal qualities that will help him as he recovers from this set-back and returns to the game.”