The entrepreneur's G1 Group aims to return the former Waverley Cinema in Moss Side Road, Shawlands – currently Tusk nightclub – to use as a picture house.
Mr King transformed the site into Tusk and Waverley Tea Rooms almost a decade ago. Possible plans include three 100-seat cinema screens, as well as an events space.
G1 already runs the Grosvenor Cinema in the west end. Director of operations David McDowall said: "I was kindly invited to attend a Shawlands Community Council meeting and took the opportunity to present the early stage plans of our Shawlands proposal.
"Our assessment of the viability of the development is still at an embryonic stage, however we are enthused by the opportunity and look forward to exploring it further with the support and input of key local stakeholders."
Local councillor David said residents would welcome a cinema in the area. He said: "A cinema would be a great boost for the area.
"This is the biggest anchor to come into the area for a long time and it will make a massive difference. It will also help boost existing local businesses."
No formal planning application has been submitted but Mr Meikle believes it could open early next year: "The cinema could be up and running in time for the Commonwealth Games."
However, plans for a cinema at Silverburn Shopping Centre, part of a £20 million expansion revealed by our sister paper The Evening Times on Saturday, could provide competition for a Shawlands cinema.