Raham Saleem and his sister Sabam were hit by a Vauxhall Astra in Brudenell Road in the Hyde Park area yesterday.
Police said Sabam had taken her brother to a shop and they were heading home when they were hit near the Makkah Masjid mosque.
Both are in a critical condition after surgery at Leeds General Infirmary, the BBC reported.
Police said the driver slowed down to see what had happened and then sped away.
Detective Supt Paul Taylor, of the homicide and major inquiry team who is leading the investigation, said: "They went to the shop to buy some onions for their mum, preparing for Eid celebrations today.
"They were crossing the road and they were struck by a vehicle, both were injured and underwent operations overnight.
"We are very concerned about their condition."
Taylor added: "Her parents have told us that Sabam in particular couldn't wait to celebrate Eid today and instead she is now fighting for her life in a hospital bed.
He said CCTV had been examined and experts would study more footage to try to identify the car's registration number.
"I would appeal to that driver to come forward so he or she can tell us what happened," he said.