The chairman of a backbench council watchdog committee has accused Labour leader Sir Albert Bore of leaving unanswered questions over the multi-million pound IT contract.
And Coun Bore, who is following the shock could be called back to the committee to answer further questions.
It follows Coun Bore's appearance before the cross-party Corporate Resources Scrutiny committee in September in which he was asked about publishing more details of the contract.
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Under the controversial contract, revenue collection and other services - but many councillors have complained that, despite cuts, it is too expensive and money could be saved and diverted to other public services.
In a leaked letter, committee chairman Coun Waseem Zaffar (Lab East Handsworth and Lozells) points out that the chief executive of Capita agreed to the principle of transparency on public sector contracts at a Parliamentary Public Accounts Committee hearing so they want Birmingham to do the same.
He adds there is also confusion around Coun Bore's replies to questions on the - particularly about the role of districts like Cannock Chase and Telford and Wrekin and the devolution deal being negotiated with Government.
A proposal document about this has been leaked into the public domain since last month's committee hearing.
Coun Zaffar, writing on behalf of members of the cross-party committee, says: "I am seeking immediate clarification on a number of issues.
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"It may be, depending upon the content of your responses and their timings, that we decide to recall you to the committee itself very soon to further clarify."
Coun Zaffar is thought to be an ally of Sir Albert Bore's leadership rival John Clancy.
Coun Bore was not available for comment on the issue.