The Education Minister has had a preliminary meeting with the SCC - a long-time advocate of the need for greater representation of parent views - and has asked it to return with a more detailed policy paper before the parental involvement Bill goes before the Scottish Parliament in autumn.
The possibility of a new parent body comes in the wake of the Scottish Executive's consultation over its draft Bill, which would abolish school boards and create a less rigid system of parent forums.
The consultation, however, has not gone smoothly, with the two existing bodies at odds over the proposals. The Scottish School Board Association has mounted a vigorous campaign against the draft legislation, while the Scottish Parent Teacher Council has offered warm support for the more flexible format.
Speculation is mounting that the Executive is preparing to amend key areas of the draft Bill before the next stage of the legislative process.
However, Mr Peacock's apparent enthusiasm for a national body with the sole purpose of representing parent views - a proposal the SCC has been pushing for more than five years, so far to little avail - may indicate that he is seeking a third way that would meet his desire to engage more with parents but would avoid having to favour either the SSBA or the SPTC over the other.
A spokeswoman for the Executive said the minister was interested in funding a national body but that it would depend on the specifics of what the SCC came up with. Discussions were still at an early stage, but Mr Peacock thought the plans were "interesting".
The SCC is due to meet senior members of the SSBA next month to discuss its proposals. Its view is that both the SSBA and SPTC are partnership bodies, whose members include significant numbers of teachers, and that they cannot therefore offer a truly independent voice for parents.
For its part, the SCC would like a national organisation to have direct links to the proposed parent forums and for this to be put in statute through the Scottish Schools (Parental Involvement) Bill.
It argues that there has been too much talk about the relationship between the school and individual families and not enough attention has been paid to how parent views as a whole can be represented. A spokesperson said:
"While partnership organisations exist to bring together parents and teachers at school and national level, there are no bodies that exist solely to represent the parental interest in education. The SCC believes this to be a significant gap in current policy."
Jennifer Wallace, policy manager for education at the SCC, said the council recognised that to be fully funded by the Executive and also be independent was "quite a balancing act" and that ultimately the SCC might favour a mixed funding approach rather than relying on a single source.
The SCC has yet to reach a conclusion on the kind of model a national parent body might adopt, but there are parallels with the health sector.
"There is no suggestion that the SCC would represent parents. People have assumed that we would expect ourselves to do it, but that is not really our role," Ms Wallace said.
Caroline Vass, president of the SSBA, said: "Parents, through school boards, already have a national body which represents their views. It is called the SSBA. Just as boards were willing to evolve, so was the SSBA. We have talked about our willingness to take forward Mr Peacock's vision, but this (the SCC proposal) seems to me to be a similar situation to the boards and forums debate - he is starting again rather than tweaking the best we have."
Judith Gillespie, development manager of the Scottish Parent Teacher Council, said she thought that the SCC model was "flawed". "What people want is something familiar to them - not something which is like the Citizens Advice Bureaux," Mrs Gillespie said.