The Kelpies, on the banks of the Forth and Clyde canal near Falkirk, have attracted thousands of visitors since opening in April 2014.
The 30m-high steel horse’s head sculptures were designed by Andy Scott, who retains their intellectual property rights (IPR).
But local councillors have called for a report to scrutinise the legal position of the present agreement in a bid to explore ways of gaining more control over marketing rights.
Members of Falkirk Council believe that as the Kelpies were paid for by a mixture of taxpayer and Lottery money, the authorities which run the attraction should have a bigger say on how they are operated.
The move follows a decision last year not to allow a giant Falkirk Football Club scarf to be temporarily tied around one of the sculptures to mark the team’s appearance in the Scottish Cup final.
The artist felt the scarf plan was “inappropriate”, a decision he said was agreed by both the council and Scottish Canals. The Kelpies were instead lit up gold for the final in May last year.
Councillor Craig R Martin wants the local authority to have more control over the Kelpies to create new revenue streams.
He said: “I am glad progress has recently been made and that Andy is working with us, but I do feel we should have more control over the Kelpies.
“They were paid for with public money to help the Falkirk area prosper and we want to maximise opportunities and encourage community groups to use the attraction in a way that respects Andy’s intellectual rights.”
Councillors David Alexander questioned the original agreement asking why the public money did not ensure control over the attraction.
Mr Alexander said: “You have got to question the arrangements in place at the start when the funds were secured.
“The Kelpies are for the people and should belong to the people of Falkirk – they are not a private art collection.”
A spokesperson for Falkirk Council said: “Scottish Canals and the council worked together on the Helix project and agreed a commissioning agreement with Andy Scott Public Art Ltd for the Kelpies.
“As is usual in such an agreement, the intellectual property rights, including copyright, remained with the creator of the Kelpies, Andy Scott Public Art Ltd, however, the company must work with the council and Scottish Canals to use this for merchandising purposes.”
The spokesperson added that the merchandising deal currently in place is there to ensure goods are “of the highest standard and in keeping with the reputation and image of the Kelpies” and ends in November 2016 when it will be reviewed.”
Mr Scott said he and the local authorities aim to promote the Kelpies as world-class artworks to a local, national and “increasingly global audience”.
He added: “We are taking considerable effort to ensure that those products are in keeping with the Kelpies’ kudos and stature as national icons and that where possible, products are sourced from local or Scottish companies. We are surprised to find out that somebody is complaining about this.
“As the artist who dreamt up the sculptures ten years ago and the person who is arguably closer to them than anyone, I am immensely proud of what they have achieved for the town and surroundings and the honours and accolades they’ve received.
“I’m deeply honoured that the people of the Falkirk and Grangemouth area have taken them so much to heart.”