Mr Hendry said: "At the moment we have litter wardens and traffic wardens.
"If a traffic warden sees someone dropping litter or a litter warden sees someone parked on a double yellow line they can't take action.
"It makes sense for them to have the powers to enforce both.
"Chewing gum on streets is a serious problem and I would fully expect anyone seen dropping gum to be given a fine. We need better enforcement."
The Evening Times told last week of one councillor who was so fed-up stepping on gum in the streets he suggested banning it altogether.
Today, Labour's Paul Carey again called for a "permanent solution" to the problem.
He said: "I have confirmation that Glasgow City Council's land and environmental services spent £65,000 on chewing gum removal in the last year.
"But it continues to cause a mess on the pavements, people's shoes and prams.
"I feel there must be a permanent solution."
But council leader Gordon Matheson, responding to Mr Hendry's comments, said: "Councillor Hendry tried to block the introduction of additional enforcement officers in last year's budget.
"Once he explains that, I will be interested to hear his ideas."
Meanwhile, anti-litter campaigners have called for those people who drop gum in Glasgow streets to "chew it, wrap it, bin it".
Carole Noble, Keep Scotland Beautiful's head of operations, said the group – which is now working alongside chewing gum manufacturer Wrigley's – wants to see special wraps sold as part of a packet or individually to help tackle the mess.
Ms Noble added: "The gum is a disgusting mess, while the material it is made of will leave a stain, even after the gum has gone.
"Millions of pounds has been invested in towns and cities with the aim of encouraging more people on to the high streets but, sadly, people continue to drop gum and litter."
She said chewing gum is one of the main gripes from the public, when questioned about litter, along with dog dirt.
Ms Noble added: "Chewing gum wraps would minimise the mess but we are depending on people having the good sense to use them.
"Our message remains – chew it, wrap it, bin it."