The Islamic State (IS) group overran the city dubbed the "Pearl of the Desert" last May, and it has since blown up UNESCO-listed temples and looted relics dating back thousands of years.
Its recapture would be a strategic as well as symbolic victory for President Bashar al-Assad, since whoever controls it also controls the vast desert extending from central Syria to the Iraqi border, experts say.
Loyalists backed by Russian air strikes were "800 metres (yards) from Palmyra" and now control areas linking it to Damascus and Syria's third city Homs, a Syrian security source said.
Rami Abdel Rahman of the Syrian Observatory for Human Rights said earlier Wednesday that regime forces were two kilometres (little more than one mile) south of Palmyra and five kilometres southwest of the city.
In Geneva, meanwhile, negotiators were making a fresh bid for a breakthrough ahead of a planned pause in the peace talks starting Thursday.
There is some hope that high-level US-Russian meetings this week could deliver the momentum needed to move on to a new round.
With the indirect negotiations in Geneva proving to be sluggish, all eyes are on Moscow since the two powers hold significant sway over the opposing sides in Syria's devastating conflict.
"The diplomatic process in Geneva is interconnected with what is taking place in Moscow," said the High Negotiations Committee, the main opposition umbrella group.
The HNC said it hoped that after the Kremlin talks "a clear message will be sent to Bashar al-Assad: He cannot continue to paralyse the political transition that the Syrian people are demanding.
