The Imagination Library was launched in 2007 as the first arm of a project pioneered in the US.
It encourages children under five to read by sending them a free book every month, with more than 80% of the town's under fives signed up to the project.
The council said the scheme was facing closure because of budget cuts.
Since 2008 over 28,000 children have joined the library, with the current membership standing at 13,000 children.
The authority said it cost Â£387,000 a year to run the service and it could not longer afford it because it had to make savings of Â£23m in the next financial year.
A final decision will be made after the month-long consultation.
If closure is agreed, the council said the library would stop in December.
People's views are wanted on plans to axe a children's reading scheme in Rotherham which was backed by US country singer Dolly Parton.