Mr Matheson said his candidature would offer a “fresh perspective” for the party due to his status as a councillor rather than an MSP.
The rule change to allow councillors to stand for the deputy leadership post was part of a package of reforms pushed through the party’s ruling body by Jim Murphy in his last act as Scottish Labour leader.
Inverclyde council leader Stephen McCabe, Scottish Labour finance spokeswoman Jackie Baillie and Labour MSP Alex Rowley - a former election agent for Gordon Brown - are also understood to have expressed an interest in the post.
Mr Matheson said: “The Scottish Labour party needs to stop feeling sorry for itself and instead focus on the concerns of the people of Scotland.
“I offer the distinct combination of experience and change. I have successfully led Scotland’s largest city for over five years.
“But because I’m not an MSP I offer a fresh perspective on the national political scene.
“It is when times are tough that mettle is shown. I am under no illusions as to the challenges ahead for the Scottish Labour Party but I’m up for the fight because I believe in Labour values.
“I welcome the opportunity of the leadership campaign to discuss with party members and wider society the changes the Labour Party needs to make if we are to regain trust.
“I hope to serve the people of Glasgow and Scotland in this new role, and support the next Leader of the Scottish Labour party.”
Labour MSP Richard Baker, who was a director of the anti-independence Better Together campaign, and Cowdenbeath MSP Alex Rowley - a former election agent for Gordon Brown - have also formally declared as candidates for the deputy leadership contest.