Unite says it has about 1,900 members at the UK's busiest container port, including the 116 engineers.
While it is only the tyre fitting operation which faces being transferred, it fears could be the start of an unwelcome trend, it says.
It has announced today that it will also start balloting its remaining 1,800 members at the docks for strike action and industrial action short of a strike on the outsourcing issue. The ballot will open on December 24 and closes on January 14.
Unite regional officer Neal Evans said: "What we have here, if this outsourcing proposal involving Universal Tyres goes ahead, is the thin end of the wedge which, we believe, will see a salami-slicing of pay and conditions in the future. This is completely unacceptable.
"At present, the threat is just to the tyre fitting operation but we fear that this could affect more of our engineering members going forward."
He added: "The crux of this dispute is that our engineering members consider that the identity of their employer is a condition of their employment contract and they do not want that being changed unilaterally.
"Concern is also snowballing amongst the rest of our 1,800 membership that they could be next for the outsourcing 'chop' - and that's why we are holding a further ballot for those members for industrial action, including strike action.
"This latest ballot is a really powerful shot across the company's bows - however, there is still time for management to row back on these flawed plans that have nothing to do with the smooth running of the port, but are framed to squeeze out the last penny of profit at the expense of the dedicated workforce."