The group, including MSPs Jenny Marra and Mark Griffin, along with councillors Scott Arthur and Monique Adams have called for an emergency meeting of the SEC to be held on Saturday to discuss the issue.
But the move has been branded "factionalism" by Mr Leonard who has refused internal calls for him to quit over the past week amid plunging poll ratings.
A letter obtained by the Scotsman to Labour general Secretary Michael Sharpe calls for the emergency meeting to be convened at the "earliest opportunity."
The motion being proposed by the group states: "The SEC expresses that it has no confidence in the Leader of the Scottish Labour Party."
The letter to Mr Sharpe states: "We are requesting an emergency meeting of the SEC to discuss a no confidence motion in the Leader of the Scottish Labour Party.
It adds: "As per the standing orders of the SEC, section 3 paragraph 2, this requires a meeting to be convened at the earliest opportunity, with our motion as the sole agenda item.
“As there is a meeting scheduled already for Saturday 12 September, we would suggest that this emergency meeting takes place on Saturday, in place of the planned meeting. We have copied the Governance and Legal Unit into this email to ensure the rules are properly followed."
The letter has also been signed by Councillor Maureen Devline, Baroness Meta Ramsay, James Adams, Jackie Martin, Jane Prinsley and Suzan King.
But the move met with a stinging response from Mr Leonard.
"The last thing Labour supporters and those we need to win back want to see is a party turning in on itself," he said.
"A lot of these people are not mandated by their organisations to vote in this way.
"That exposes how this is motivated by faction and that this is an entirely factional move
"They have nothing to say or offer, no ideas, no plan to protect jobs and living standards against the backdrop of the Covid crisis.
"That is what I want us to concentrate on: the real struggle the people of Scotland are in."
The SEC comprises MSPs, councillors, trade union representatives and other affiliated representatives.
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