Plans are for the issue to be on the November 2014 general election ballot. Fallin is expected to be on the ballot to seek a second term.
Supporters said they had no estimate of how much the petition drive and advertising would cost, nor have they identified potential donors.
It's also unclear whether the proposal would change state law or the state constitution. What type of change is sought determines the number of signatures of registered voters required to put an issue on the ballot.
A statutory change would require 8 percent of the 1,034,767 votes cast in the 2010 gubernatorial election, or 82,782 signatures; a constitutional change requires 15 percent, or 155,216 signatures.
Fallin was flanked by University of Oklahoma medical students wearing white jackets, along with health officials and representatives of several cities.
Oklahoma has the fourth-highest smoking rate in the nation, Fallin said. About 6,000 people die each year in the state from smoking-related causes.
About 700 of those die from secondhand smoke.
“These are unnecessary deaths, and they're heartbreaking,” the governor said.
Fallin said both her parents smoked and died from smoking-related causes. Her mother suffered three heart attacks and several strokes before she died in 2005 at the age of 78.
“It's a personal issue for me,” she said.