Coun Box, who has led the council since 1998, scraped through by a small margin of votes at a party meeting on Monday.
He has come under pressure this year following damning revelations about Wakefield's children's services.
An Ofsted report in July said that there were "serious and widespread" failures in the department, with the district's vulnerable young people left at risk as a result.
At what was described as a "tense" meeting by one source, a motion was put forward calling on Coun Box to resign.
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It is understood a vote was conducted by a show of hands without debate, rather than a secret ballot, and the motion was defeated. Reports of the margin of victory have varied between three and five votes.
One unamed councillor, who voted in favour of the motion, said they believed the Labour leader had been "mortally wounded" by the vote.
They said: "I don't think he can survive with 20 plus members of the council having no confidence in him.
"As a council we should hang our heads in shame about what's happened with children's services.
"There's been a loss of direction at the top."
Another councillor said that the meeting was "remarkably civil under the circumstances".
They said: "I was going along expecting a shouting match, but it wasn't like that in the end.
"It has arisen indirectly because of children's services, which is a major problem.
"I'd be surprised if this is the end of the matter."
Another source, who supported Coun Box, said: "There's differences of opinion within the group, and there's a few reasons why people want him to go.
"But I can see no reason why he shouldn't carry on. At the end he asked us to continue our work and everyone was united."
Coun Box is expected to attend the first full council meeting following the summer recess on Wednesday afternoon.
Labour's chief whip, Coun Richard Forster said: "At today’s meeting of the Labour Group, the group confirmed their continued support of Councillor Peter Box in his role as Leader of the Council."
Labour said they could not confirm the exact number of votes cast on either side of the motion.