The amount could go up because the child is still being treated.
The child was fed on Sanlu baby formula from birth, Caijing magazine reported.
The Chinese government has ordered hospitals to provide free treatment to infants made ill as a result of contaminated milk products.
However, the baby at the centre of the lawsuit is being treated at Beijing Children's Hospital, which has only offered free treatment to children diagnosed ill after September 12, when the scandal broke, Caijing said.
The scandal over milk and milk products tainted with the industrial chemical melamine has sickened more than 50,000 children across China and led to the deaths of at least four babies.
The scare has seen hospitals across China besieged by crowds of anxious parents worried about their children's health.
Melamine is believed to have been deliberately added at some point in the supply chain to disguise milk that had been watered down to fool quality checks and increase profits.
If ingested the chemical can cause kidney stones and other potentially fatal complications, especially in young children.
On Wednesday China's People's Daily newspaper reported that Sanlu bosses had initially asked for government help to cover up the extent of the problem when reports of the contamination first broke.
According to the Communist Party newspaper, Sanlu wrote to government officials in on August 2 asking for help in "managing" the media response to the case.
The letter - addressed to top government officials in the city of Shijiazhuang, home to Sanlu's headquarters - was written just days before the opening of the Beijing Olympics.
"Please can the government increase control and coordination of the media, to create a good environment for the recall of the company's problem products," the People's Daily cited the letter as saying.
"This is to avoid whipping up the issue and creating a negative influence in society."