A 38-minute recording of the talks was posted online late Friday on Russian social media, with the participants discussing the possible use of German-made Taurus missiles and their potential impact.
The discussions also covered the use of long-range missiles provided to Kyiv by France and Britain.
"It is about using this recording to destabilise and unsettle us," said German Defence Minister Boris Pistorius, adding that he "hoped that Putin will not succeed".
Pistorius said he was not aware of any further leaks at the army and added that he would await the result of a military probe into the case before drawing any conclusions.
Kyiv has long been clamouring for Germany to provide it with Taurus missiles, which can reach targets up to 500 kilometres (about 300 miles) away.
Chancellor Olaf Scholz has so far refused to send the missiles, fearing that it would lead to an escalation of the conflict with a nuclear-armed Russia.
- 'Information war' -
With the war in Ukraine in its third year, Kyiv has ramped up its pleas for more military support from allies.
On the frontlines, its soldiers are outgunned and outnumbered and ammunition shortages are being felt.
The acquisition of Taurus missiles would provide a massive boost for Ukraine as Kyiv struggles to fend off Russia's invigorated push on the frontlines.
France and Britain have supplied Kyiv with SCALP or Storm Shadow missiles, both of which have a range of about 250 kilometres.
But Scholz on Monday said that Germany could not justify matching British and French moves in sending long-range missiles to Ukraine and supporting the weapon system's deployment.
"This is a very long-range weapon, and what the British and French are doing in terms of targeting and supporting targeting cannot be done in Germany," Scholz said, without specifying exactly what he meant.
Part of the conversation in the leaked recording however appeared to call into question Scholz's explanation on why he could not provide Kyiv with the missiles.
Britain has denied that it had any direct involvement in operating the missiles.
"Ukraine's use of Storm Shadow and its targeting processes are the business of the Armed Forces of Ukraine," a Ministry of Defence (MoD) spokesperson said in a statement to AFP.
Former Russian president Dmitry Medvedev, now deputy head of the Security Council, on Sunday said: "Germany is preparing for war with Russia".
Speaking at a diplomatic forum in Turkey on Saturday, Russian Foreign Minister Sergei Lavrov said the recording indicated that Ukraine and its backers "do not want to change their course at all, and want to inflict a strategic defeat on Russia on the battlefield".
With politicians in Germany urging answers over the wiretap, Pistorius warned that the leak is "part of an information war that Putin is carrying out".
"We should not fall for Putin's line," he said.
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