Ukrainian media reported that President Volodymyr Zelenskiy asked General Zaluzhnyi, head of the Ukrainian army, to step aside this week, but that he refused.
His exit as army chief would represent a huge shake-up in the Ukrainian military, which is fending off multiple Russian attacks in the east as uncertainty grows over the future of vital U.S. and European Union support.
The source, who requested anonymity to discuss sensitive matters, was unable to say exactly how or when the job offer was communicated to Syrskyi.
The Ukrainian General Staff and president's office did not immediately respond to requests for comment.
Zaluzhnyi is seen as a hero to many in Ukraine after his troops successfully defended Kyiv against Moscow's forces at the start of Russia's full-scale invasion nearly two years ago.
Ukrainian media have tipped Syrskyi and military spy chief Kyrylo Budanov as two possible successors to Zaluzhnyi.
Under Zaluzhnyi's command, Ukraine defied overwhelming odds to beat back Moscow's initial offensive and then regained swathes of occupied territory later in 2022.
The defence of Kyiv and a lightning advance in the northeastern Kharkiv region were overseen by Syrskyi.
But Ukraine's much-vaunted 2023 counteroffensive failed to make serious inroads into heavily-defended Russian positions in the occupied south and east.
It was not immediately clear how Kyiv's allies in the West would view moves to replace Zaluzhnyi, who retains overwhelming popularity in Ukraine.
The Kremlin said speculation about Zaluzhnyi's future reflects the problems Kyiv faces nearly two years into the war.
(Reporting by Dan Peleschuk; writing by Tom Balmforth; editing by Ros Russell)
