The messages, apparently sent by supporters of the union's hard-left general secretary, repeated classic anti-Jewish tropes including accusing Mr Coyne of having the "shadowy backers" of a "Jewish Mafia".
Another Twitter post claimed he was "cosying up to powerful Jewish conservative supporters of Zionism and ready to sell his soul".
Mr Coyne was said to be considering reporting the string of antisemitic insults to police for investigation as a possible hate crime on Saturday evening.
One tweet, written by a user named @1UnionBloke, and also including links to the official Unite Twitter feed, included the words: "Be warned @gerard_coyne has backing now!! @UniteforLen."
Others messages also sent to Mr Coyne from the same Twitter feed included one which read: "His anti anybody who isn't Jewish is exposed. @gerard_coyne also reveals his shadowy backers. So @UniteforLen."
A source said Mr Coyne was "upset" and "disgusted" at the nature of the messages - and was considering his next step.
There was widespread condemnation of the antisemitic nature of the Twitter messages addressed to Mr Coyne.
Observer journalist and author Nick Cohen wrote: "Len McCluskey's supporters are disgusting. There's no racist propaganda too disgusting for them to use against Gerard Coyne."
In the JC interview, Mr Coyne had called for Jews, and members of other faiths, to rejoin Unite if he became leader after the forthcoming ballot of the union's 1.42 million members.
Mr Coyne, who is currently the union's West Midlands head, also attacked the involvement of three Unite representatives on Labour's NEC who argued against issuing warnings to two members of the Oxford University Labour Club following an investigation into alleged antisemitism and bullying claims.
The ballot for the Unite election opens on March 27 - with the result of the election announced at the end of April.
