Mr Klitschko derided their disputes as "senseless" amid Russia's military campaign.
He said 430 "warming centres" were helping residents to cope with the effects of Russian attacks on power stations and that more than 100 more were planned in case of extreme conditions.
Mr Zelenskyy said last week that there were many complaints about measures adopted in the capital.
"I do not want to become involved in political battles, particularly in the current situation," Mr Klitschko said in a video posted on Telegram.
"That is senseless. I have things to do in the city."
He had been involved in disputes with Mr Zelenskyy before the invasion, and on Sunday said the president's allies had engaged in "manipulation" of the city's efforts, including "incomprehensible photos" posted online.
"To put it mildly, this is not nice — not for Ukrainians or for our foreign partners," Mr Klitschko said.
"Today, like never before, everyone must unite and work together. And here we have some sort of political games."
But one of Mr Zelenskyy's top allies landed a new blow, saying Mr Klitschko's explanations "differ considerably" from what residents were reporting.
David Arakhamia, head of the president's Servant of the People group in Parliament, said on Telegram that the mayor had a week to rectify issues and submit to inspections of the help centres.
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"I have a proposal," Mr Arakhamia said. "City authorities have a week to correct the errors and we conduct inspections together with the mayor."
Mr Zelenskyy has in the past week promoted the efforts of thousands of "invincibility centres" set up throughout Ukraine to provide heat, water, internet and mobile phone links.
In his nightly video address on Friday, he said Mr Klitschko had not done enough to help beleaguered residents.
"To put it mildly, more work is needed," Mr Zelenskyy said.
The president was elected in a landslide in 2019, five years after Mr Klitschko first took office.
Before the war, the two leaders had clashed over the way the capital and its services were being run.