North Korean leader called Mr Trump a "dotard" and a "frightened dog" over the threats of annihilation the US President made during a speech at the United Nations General Assembly.
Mr Trump responded the next morning by calling Mr Kim a "madman" and threatening to test him "like never before".
Moscow's Sergei Lavrov described the leaders as "hotheads" who needed to "calm down".
"Yes, it's unacceptable to silently watch North Korea's nuclear military adventures, but it is also unacceptable to unleash war on the Korean Peninsula," he said.
Calling for a political solution to the dispute, he said: "Together with China we'll continue to strive for a reasonable approach and not an emotional one like when children in a kindergarten start fighting and no-one can stop them."
Mr Lavrov told a news conference there were many people “who would like to try not to pursue” military action and sanctions, but rather try peaceful solution.
He added that discussion between the nations could be meditated by a "neutral" European country.
He said he had no new initiatives to bring the two sides together, saying he believed “the potential” for the Russian-Chinese freeze-for-freeze proposal “is not yet exhausted".
Such an agreement would halt North Korean nuclear and missile tests in exchange for the US and South Korea stopping their joint military exercises. Mr Trump's administration has rejected it.
Tensions have racheted up between America and North Korea after the isolated state carried out a series of missile tests, two of which flew over US-ally Japan.
Mr Trump used a speech at the UN to say that the US would "totally destroy" the communist nation if it was forced to defend itself or one of its allies.
Mr Kim responded with an unprecedented personal statement where he said Mr Trump would "pay dearly for his speech". Experts have said the speech carried added weight because it was unusually addressed to an international audience.
Mr Kim said Mr Trump had insulted his country and he threatened to "surely and definitely tame the mentally deranged US dotard with fire".