Wrexham council said it estimated that new rules will land it with a £700,000 annual bill.
The authority said adopting the new Welsh government standards would lead to cuts elsewhere.
The Welsh government said the Welsh Language Commissioner would decide which standards councils had to meet.
Other councils have told BBC Radio Cymru's Post Cyntaf they also faced extra costs, though some local authorities said the rules will not have a financial impact.
Language Commissioner Meri Huws said it would not be appropriate to make any comment until she had time to consider councils' submissions.
Wrexham council has written to Ms Huws to say it could meet 134 of 164 standards but that it would cost £700,000 extra each year to adhere to the remaining standards against a backdrop of £45m in budget savings over the next three years.
Councillor Hugh Jones, Wrexham's lead member for communities and partnerships, said the authority would be "forced to make budgetary savings" elsewhere to adopt the remaining standards which "place at risk" its ability to support library and heritage services as well as the Stwit theatre in Rhosllannerchrugog.
Pressure group Dyfodol i'r Iaith (Future For The Language) said while extra standards were needed, they were too "bureaucratic".
Adopting new Welsh language standards could cost some councils hundreds of thousands of pounds every year, it is being claimed.