Nearly 25,000 signatures - 10% of Cardiff voters - are needed to trigger a Yes/No referendum on the issue.
Cardiff North Tory MP Craig Williams and former Labour council cabinet member Ashley Govier are among those backing the plan.
There are 18 elected mayors in England with more to come, but none in Wales.
Mr Williams said: "If Cardiff doesn't keep up with the other big urban centres across the UK and the world then we won't be in a position of strength either economically or politically, and that worries me."
But Cardiff council's Liberal Democrat group leader Judith Woodman told BBC Wales there were questions over the cost of having a mayor, and how they would be held accountable.
"It's promoting one person who has too much power," she said.
If enough signatures are gathered, a referendum could be held in autumn 2016.
Depending on the result, a mayoral election could take place in 2017.
Ceredigion is the only Welsh local authority to have had a referendum on the matter.
Voters there rejected the idea in 2004 by a margin of nearly three to one.
A directly elected mayor for Cardiff could help the city compete on the global stage, campaigners claim, as they launch a petition on the idea.