It would mean some schools being taken from council control and being converted to academies.
There are four investigations examining "Trojan Horse" claims that groups of hardline Muslims have been trying to take over schools in the city.
A government source says that no options have been ruled in or out.
The trust which runs Park View, one of the schools caught up in the claims, issued a statement on Thursday dismissing allegations of extremism as false.
The education watchdog Ofsted and the Department for Education are expected to deliver reports in June on the next steps for schools which have faced these allegations.
It is understood that ministers have approached heads of successful schools and academy trusts about taking over other schools where there have been concerns.
This could include taking schools away from local authority control, converting them into academies - directly funded by central government - and putting them under the leadership of high-achieving head teachers.
Groups of outstanding schools like the Perry Beeches chain in Birmingham may have a role.
As some of the schools caught up in the Trojan Horse claims are already academies, it could mean schools being reassigned to other academy trusts.
Birmingham City Council said it would not be appropriate to comment on any "proposed action" until Ofsted's inspection reports had been published.
"We can say, however, that we are in regular contact with DfE and this action has not been discussed with us," said a council spokesperson.
The GMB union warned: "It is absolutely disgusting that the government is pre-empting the outcome of the investigation into these schools and planning to sell these schools on to their friends in the academy chains."
Before the Trojan Horse claims, Ofsted chief Sir Michael Wilshaw had spoken publicly of his belief that the local authority might need to be broken up.
Birmingham is the biggest local authority in England - and it is believed that private soundings have shown it would be difficult to find a private sector contractor willing or able to take on the running of such a wide range of services.
But another option would be for some or all of the council's education and children's services to be hived off to an independent trust.
This model has been used before where central government believed there was a need for radical intervention - such as in Hackney in London.
Birmingham City Council's chief executive, Mark Rogers, was recorded at a meeting with school representatives last week, where he warned of a "firestorm" when the Ofsted reports were published, and telling his audience to prepare for "significant structural changes" within the education department.
The investigations into extremism in Birmingham schools had been sparked by an anonymous and unverified letter claiming that there was a "Trojan Horse" conspiracy by a group wanting to impose a more hardline Muslim agenda on schools in the city.
This included claims that head teachers were being replaced with staff who were more likely to be sympathetic to these religious beliefs.
The authenticity of the letter has been disputed, but the National Association of Head Teachers says there seem to have been "concerted efforts" to take over six schools in Birmingham.
However, claims of extremism have been categorically denied by Tahir Alam, chairman of governors at Park View School.
And the Park View Educational Trust issued a statement arguing that the response to the letter had created a climate which encouraged false allegations.
The claims about religious extremism were driven by "the settling of political scores" and disgruntled former staff, the trust argues.
"Our assemblies, or acts of worship, are used to focus on the spiritual and moral development of pupils and are relevant to their school and home lives," said the statement from Park View.
"This includes looking at the common values of kindness, mercy and forgiveness and talking to students about studying hard, and being a good friend."
Super-heads could take over running clusters of schools in Birmingham, as the government considers its response to allegations of extremism.