Dave Prentis, the general secretary of Britain's largest trade union, confirmed that UNISON is taking "urgent legal advice" and is considering making a complaint to the police.
"Clarkson's comments on The One Show were totally outrageous, and they cannot be tolerated," Prentis said.
"We are seeking urgent legal advice about what further action we can take against him and the BBC, and whether or not his comments should be referred to the police."
He continued: "Public sector workers and their families are utterly shocked by Jeremy Clarkson's revolting comments. We know that many other licence fee payers share our concerns about his outrageous views.
"The One Show is broadcast at a time when children are watching - they could have been scared and upset by his aggressive statements.
"An apology is not enough - we are calling on the BBC to sack Jeremy Clarkson immediately. Such disgusting statements have no place on our TV screens."
Prentis added: "Jeremy Clarkson clearly needs a reminder of just who he is talking about when he calls for public sector workers to be shot in front of their families.
"While he is driving round in fast cars for a living, public sector workers are busy holding our society together - they save others' lives on a daily basis, they care for the sick, the vulnerable, the elderly.
"They wipe bottoms, noses, they help children to learn, and empty bins - they deserve all our thanks - certainly not the unbelievable level of abuse he threw at them."
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Watch Jeremy Clarkson's comments on The One Show below: