AMMAN (Reuters) - Syrian rebels said on Thursday they had started pushing into Hama, a major city where pro-government forces backed by intense Russian air strikes are trying to stave off a new rebel victory and halt the insurgents' lightning advance.

Rebel commander Hassan Abdul Ghany posted on social media that the insurgents had begun to penetrate Hama. State media had earlier reported that pro-government forces had repelled an attack.

Insurgents have been battling to try to enter Hama since Tuesday and there was heavy fighting overnight with the Syrian army and allied Iran-backed militia groups supported by a Russian bombardment, both sides said.

The rebels took the main northern city Aleppo last week and have since pushed south from their enclave in northwest Syria, reaching a strategic hill just north of Hama on Tuesday and advancing towards the city's east and west flanks on Wednesday.

Hama has stayed in government hands throughout the civil war, which erupted in 2011 as a rebellion against President Bashar al-Assad. Its fall to a revived insurgency would send shockwaves through Damascus and its Russian and Iranian allies.

The city lies more than a third of the way from Aleppo to Damascus and its capture would open the road to a rebel advance on Homs, the main central city that functions as a crossroads connecting Syria's most populous regions.

Hama is also critical to the control of two major towns with big minority religious communities, Muhrada which is home to many Christians and Salamiya where there are many Ismaili Muslims.

The most powerful rebel faction is the militant Sunni Islamist Hayat Tahrir al-Sham (HTS), the former al Qaeda affiliate in Syria. Its leader Abu Mohammed al-Golani has pledged to protect Syria's religious minorities but many remain fearful of the insurgents.

On Wednesday Golani visited Aleppo's historic citadel, a symbolic moment for rebels who were driven out of the city in 2016 after months of siege and intense fighting, their biggest defeat of the war. Aleppo was Syria's biggest city before the war.

HTS and the other rebel groups are trying to consolidate their rule in Aleppo, bringing it under the administration of the so-called Salvation Government they established in their northwestern enclave.

Aleppo residents have said there are shortages of bread and fuel, and that telecoms services have also been cut.

Russia and Iran were crucial to Assad's success in recovering most Syrian territory and all main cities from 2015-20, and they have sworn to help him again.

But Moscow has been focused on the war in Ukraine, while Iran's most important regional ally Hezbollah, which for years played a critical role shoring up Assad in Syria, has suffered heavy losses to Israel in Lebanon over the past two months.

Russian air strikes across rebel-held areas of northern Syria have sharply intensified over the past week. Iran-backed militia groups from Iraq reinforced front lines after bringing fighters across the border on Monday, Iraqi and Syrian sources said.