Hillingdon, where Heathrow airport is located, says the number of children seeking asylum entering the area has risen by nearly 250% in the past 11 months, outstripping government grants.
Hillingdon says it now faces a funding gap of between£2m and£3m, which administration leader Richard Barnes estimates could add around four percentage points to the borough's planned increase of 7.5-8%.
Mr Barnes said: 'We have no choice if we don't get a grant from central government.'
The county has identified a£1.7m budget shortfall, caused partly by the knock-on costs of recruiting carers for child asylum seekers. It says that without extra money it will raise its projected tax by just under 0.7 of a point to 8.4%.
Kent leader Sandy Bruce-Lockhart said: 'It is not right for Kent council tax payers to have to pay for something that is agreed to be a national issue.'
The Local Government Association says asylum seekers have cost council tax payers in London and Kent around£30m in the current financial year.
The flow of refugees has been exacerbated by the Kosovo conflict, leading to a leap of more than 50% in the number of asylum applicants in 1999.
But despite these figures, and reports suggesting councils across the country may be about to follow Kent and Hillingdon, no other council has yet indicated it will also raise council tax.
LGA asylum seekers team leader Mike Boyle said: 'Clearly Kent has got a problem, Hillingdon has got a problem. They're having to incur costs they shouldn't be incurring. Our position is that the government should reimburse this in full.'