Some 22 out of 38 council members had signed the motion after friction resulted by Wigneswaran's move to sack some ministers of his administration.
Two of the sacked ministers - Agriculture Minister P Aingaranesan and Education Minister T Kurukularaja - were subject to a corruption probe.
A committee appointed by the chief minister last year submitted its report last week. It found evidence on the charges made against the two ministers.
He and his Tamil National Alliance (TNA) have been at loggerheads since early 2015. The TNA's moderate faction has been backing the government's reconciliation programme.
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However, Wigneswaran has expressed dissatisfaction with the government's delivery of reconciliation pledges.
Cooray's office said he may send a letter to Wigneswaran asking him to prove his majority in the house.
Sources said that the motion also seeks appointment of CVK Sivagnanam, the current council Chairman, as the new chief minister.
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