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The demand was made by Cosatu general secretary Zwelinzima Vavi who announced on Monday morning that Cosatu would appoint new trustees to the board of the JCT.
Cosatu has the right to appoint up to eight trustees, while the Federation of Unions of SA and the National Council of Trade Unions appoint two each.
However, the trust is an independent legal entity governed by a trust deed.
At the last JCT board meeting in April, Madisha received unanimous support from the trustees present.
These included Cosatu appointees Amon Ntuli, formerly of the clothing and textile union, Sactwu, and academic Eddie Webster.
The initial aim of the trust was to include representatives of the three main labour federations as well as of civil society.
Among the first trustees were the late Beyers Naude and the late chief rabbi, Cyril Harris, both appointed by Cosatu.
Another still current trustee is Molefe Tsele, who is now South Africa's ambassador in the Democratic Republic of Congo and is generally not available.
Over the years, the JCT has written to Cosatu on several occasions asking that vacant posts be filled.
"Until now, there has been no response. I only hope that this does not mean Cosatu wants to remove the three regular trustees, Prof Webster, Amon Ntuli and Petrus Mashishi [president of the municipal workers' union Samwu]," said Joseph Maqhekeni, president of the National Council of Trade Unions.
"This trust has operated efficiently and well. It should not be disrupted because it is a trust that belongs to all South Africans."
Sapa