MISRATA, Libya (Reuters) - Libyan forces allied with the U.N.-backed government have been shelling and carrying out air strikes on the center of Sirte city in a siege of Islamic State militants there, an official said on Tuesday.

Militants defending Islamic State's last stronghold in Libya have been keeping Libyan forces back with sniper fire and mortars in Sirte where they are now surrounded after a two month campaign to take the city.

The fall of Sirte would be a major blow to Islamic State, which took over the city a year ago in the chaos of a civil war between rival factions who once battled Muammar Gaddafi in 2011.

"Our forces have...targeted militants with artillery and air force around Ouagadougou complex, Ghiza Asskariya district, and in the city center," said Rida Issa, spokesman for Misrata forces fighting in Sirte.

"They have targeted Islamic State members, vehicles, ammunition stores, and control rooms."

He said one Misrata fighter was killed and 20 others wounded in a mortar strike on their position in the Zaafran frontline, near the roundabout where Islamic State once crucified victims.

The bodies of around 13 Islamic State fighters were found, but Misrata forces were driven back by sniper fire.

Western powers are backing Prime Minister Fayaz Seraj's government that moved into Tripoli three months ago in an attempt to unify two rival governments and various armed factions. Seraj is working with a unified National Oil Corporation to restart the oil industry.

But while powerful brigades from Misrata city support Seraj for now and lead the fight to liberate Sirte, other hardliners to the east are still opposing him and his government has made little progress in extending its influence.

After a rapid success in driving Islamic State back from a coastal strip of territory it controlled, the battle for Sirte has slowed to street-by-street fighting as Misrata forces clear out residential areas.

Misrata commanders say a few hundreds militants are dug in around the Ouagadougou complex, the university and a city hospital. They are cautious of advancing rapidly after more than 200 fighters died in the campaign so far.

While forces from the city of Misrata are fighting Islamic State in Sirte, rival brigades allied to Gen. Khalifa Haftar's Libyan National Army are fighting to the east on another front in Benghazi and around another eastern town. Haftar's hardline backers reject Seraj's government.

(Additional reporting by Ahmed Elumami in Tripoli; Writing by Patrick Markey; editing by Ralph Boulton)