Boston mayoral candidate Bill Walczak is calling on the School Committee to continue the search for the next superintendent instead of waiting until a new mayor is elected.

Walczak said the committee should use the next few months laying the groundwork for the search, such as establishing qualities it wants in a superintendent, hiring a search firm, and selecting a list of candidates from which the new mayor can choose.

“We ought to be getting something going,’’ Walczak said. “But we ought not to be choosing or getting down to a final list. What we need to do is hit the ground running.”

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Walczak’s comments seemed directed at rival Martin J. Walsh, who has been publicly urging the committee to suspend its search. Saying the committee is rushing the process, Walsh urged a more viable and thoughtful process.

“To start the process without the input from the mayor who will lead the city during the tenure of the next superintendent makes no sense,’’ Walsh wrote in a statement on his website. “It is wrong to rush opportunities for community input during the summer.”

So far, the search to replace Carol Johnson as superintendent has mostly been quiet. In the spring, the School Committee sketched out a rough timeline for the next several months that includes holding a series of community meetings in August to hear the public’s views on what it wants in a superintendent.

The committee plans to begin the hiring process for search firm in September and October. And it will spend the rest of the year coming up with a short list of candidates to present to the new mayor, school officials said.

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