Sheffield arena was built with funds from Sheffield City Council
A South Yorkshire business has objected to the development of Leeds arena, claiming it will cost it £1m a year and lead to job losses.
Sheffield International Venues (SIV), the group behind Sheffield Arena, said government money should not be used to favour one Yorkshire city over another.
Regional development agency, Yorkshire Forward, has backed the Leeds arena but cannot confirm its level of funding.
A site near the Merrion Centre has been chosen for the £30m development.
Chief executive of SIV Steve Brailey said: "I could understand Leeds residents and the council wanting their own arena and if they decide to use council funds in this way Sheffield would have to accept it.
"But we object if Yorkshire Forward is helping to fund the project.
"They should not be funding an investment in one part of our region that will seriously damage the business of an existing venue."
The 13,000-seat Sheffield Arena was built with Sheffield City Council funds.
A spokesperson for Yorkshire Forward said: "We have given consideration to the effect Leeds arena will have on Sheffield and we have concluded that it will benefit the region as a whole, as it will support the growth of Leeds and compliment the region's cultural offering."
Yorkshire Forward said the arena plans were still being developed and at this stage it could not confirm its level of support.
Sheffield Attercliffe MP, Clive Betts, said there was "no justification" and he would be lobbying the development agency to rethink its support for the Leeds project.