Olympic heptathlon gold medallist Ennis trains at the stadium. She was also discovered at the stadium when she went to a summer holidays athletics club when she was 10.
After her triumph in August, some people called for the Don Valley Stadium to be renamed in her honour.
It is also home to the City of Sheffield Athletics Club.
On Thursday, after she received her CBE at Buckingham Palace, Ennis said: "Having that iconic stadium in my home city is incredible. And to lose that would be such a shame for future athletes coming through. So I hope that the right decision's made."
The council said it subsidises every visit by more than £5 and it requires major repair and maintenance work - totalling around £1.6 million. It has proposed the reopening of the track at the smaller Woodbourn Road Stadium nearby.