Just 18 percent of those asked said they would "definitely" get their child two jabs of the vaccine if it was approved for use in the age group.
In contrast, almost two in five parents said they would refuse to get their child immunized or only do so if it was required.
More than 1,800 parents took part in the survey by health pollster the Kaiser Family Foundation, including 181 with children younger than five years.
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It comes as the Food and Drug Administration (FDA) considers approving Moderna’s jab for children aged six months to five years. Pfizer is also expected to call for its vaccine to be used in the age group later this month.
Many scientists have become concerned about over-vaccinating children, however, because they face only a small chance of serious illness from the disease.
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Pressure has been increasing on the FDA to approve Covid vaccines for the youngest children from parts of the left-wing media.
Just last week the New York Times ran a story claiming their failure to approve the jabs for under-5s had "bred frustration" among many parents.
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However, today's survey reveals that the vast majority of parents, in fact, do not intend to get their children vaccinated against the pandemic virus.
Most parents — 38 percent — said they intended to wait until other children had got the vaccine, before deciding whether to take their own vaccine.
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Some 11 percent warned they would not take their child for the vaccine unless their state required it.
However, 27 percent said they would not get a child less than five years old the vaccine under any circumstances.
The survey was conducted between April 13 and 26, which was before Moderna requested to have its vaccine approved for use in the age group.
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There has been pressure to get Covid vaccines approved for even the youngest children.
A number of experts have raised concerns about over-vaccinating children, however, who face a small risk of becoming seriously ill with COVID and a vanishingly small chance of death.
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There are also fears over myocarditis, a form of heart inflammation detected in about one in 20,000 boys following vaccination. Girls are less at risk of complications.
While in most cases, the condition is mild, scientists are not yet sure of the long-term effects.
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