Commissioners Tim Hagan and Jimmy Dimora are ready to enact the hike next month.
Cuyahoga County already has the state's highest sales tax at 7 1/2 percent.
The boost would raise $42 million a year. The money would go to help finance a much-talked about project linking a new Convention Center with a Medical Mart Exhibit Hall to display the latest health care technology and bring visitors to Cleveland.
There will be two public hearings. The tax would take effect October 1st.
But many county residents claim it's their right to vote on the tax.
Opponents of the tax increase could force a ballot vote by the people by collecting an estimated 44,000 signatures, or 10% of those voting in the last governor's election.
The project has been under discussion for more than a year. Commissioners Hagan and Dimora say Medical Mart boosters need a public show of support.They promise to rescind the tax if the project falls through.
Negotiations are under way with Forest City. The Medical Mart would be located in the old Dillard's/Higbee's building on Public Square.
Commissioner Peter Lawson Jones supports the project, but opposes a quarter percent sales tax hike. He suggests a smaller hike and other taxes as better funding sources.
Mayor Frank Jackson has previously said he supports the Commissioners' action.