Gedling Country Park opened in March on the site of a former colliery, where the miners died.
The Friends of Gedling Country Park must win public votes to receive £25,000 from the OneFamily Foundation, which funds community projects.
Chairman Terry Lock said the garden would feature old mining artefacts.
Voting started this week for 100 projects that applied for OneFamily Foundation funding.
When voting ends on 29 June, the top four would receive £25,000 each.
"It's a beautiful place with spectacular views and it's easy to forget the dangerous work that used to go on here," Mr Lock said.
Rings that supported the original mining tunnels at the Gedling Colliery would be used to create pergola rings if the memorial garden was built, the group said.
The 240-acre Gedling Country Park cost £1.1m to convert. There were plans to create the country park since the colliery shut in 1991.
A community group is hoping to win funding for a memorial garden inside a new country park to commemorate 128 dead miners.