In England, only schools with academy status are allowed to form trusts to sponsor weaker schools.
And, the Local Government Association said, a shortage of "good quality" sponsors often left failing schools "in the dark about their future".
Ministers said there was no legal frame for council schools to be sponsors.
The Local Government Association says 91% of council maintained schools are rated  good or outstanding by education watchdog Ofsted - so they should be allowed "to play a direct role in raising education standards and improving life chances, including taking on the running of failing academies".
New analysis of Ofsted ratings for the LGA by education analysts Angel Solutions found a higher percentage of good and outstanding grades among council maintained schools than among academies.
Of 14,890 council maintained schools inspected, 91% are rated good or outstanding by Ofsted, compared with 85% of 5,058 academies, say the researchers.
There are wide variations within the academy category, with sponsored academies, which are forced into academy status after poor Ofsted ratings, predictably faring worse than converter academies, which must be rated good or outstanding before being allowed to convert.
The LGA points out that the majority of academies are converter academies.
Inspection ratings for 4,103 converter academies showed 89% rated good or outstanding but among the 955 sponsored academies inspected the figure was only 65%, according to the analysis.
Richard Watts, chairman of the LGA's Children and Young People Board, said the figures proved councils had "the track record, experience and expertise to help lift schools out of academic failure".
"The government must commit to removing the unnecessary red tape and give high performing maintained schools the option of becoming academy sponsors.
"Councils want to be regarded as improvement partners, not obstructionists to school improvement," said Mr Watts, who is also the leader of Islington Council.
"With a shortage of academy sponsors and struggling schools currently in the dark about their future the simplest remedy is to give councils the power to turn these schools around where this is the best option locally."
Mr Watts also voiced concerns about the local knowledge and capacity of the eight Regional Schools Commissioners, whose job is to oversee academic standards.
He said that without the help of councils "the early warning signs of failing" risked being overlooked.
"It is not acceptable that we have to wait for poor exam results, whistle-blowing about financial impropriety or an Ofsted inspection to trigger intervention.
"Councils are best placed to oversee school effectiveness and take immediate action where required," added Mr Watts.
A Department for Education spokesman said there was no legal framework for a council or council-run school to sponsor a school unless they opened a trust - and this would involve converting to academy status.
"We would encourage good or outstanding council-run schools to apply to become a sponsor so they can share their expertise," said the spokesman.
The spokesman said councils could not run multi-academy trusts "as legally less than 20% of members and trustees are allowed to be 'local authority influenced' to ensure the trust remains autonomous from the local authority".
A ban on top council-run schools sponsoring failing schools amounts to "red tape" and should be dropped, say council bosses.