Both clubs are keen to create a hub at the stadium that will make it at the very heart of the community.
Glencairn vice-president Alex McArthur told the Reformer one of the key aspects of the plan was gaining the support of those living nearby.
He said: “A few of the members at the meeting asked if the Burnhill residents were going to accept the plan and how they would react.
“The answer to that is, I don’t know yet, but we would like to have some dialogue with them because if it does get to a point where we are seeking planning approval, we want the support of the community.
“Both clubs want to go down the route of being community clubs. We want to be part of the area. No-one is here to make millions of pounds.”
Local resident and member of the Burnhill Action Group, Kim Jackson, believes the community will back the move but she admitted there were some concerns.
She said: “I like to think people will welcome it.
“But there are already concerns about the traffic, especially in Toryglen Road where every fortnight you can’t get parked.
“There is also the noise to think off. I can hear the tannoy from my house and it must be worse for those in Toryglen Road and Penman Avenue.
“If there is a game every Saturday or Sunday, that could be a distraction.”
The action group has already worked closely with the Glens on other projects and Kim said it was vital for both clubs to open discussions with residents. “People here are talking about this and we’ve discussed it at a couple of action group meetings.
“It looks like a good idea but it would be nice to sit down with them and put our concerns before they press on.
“People here will support it as long as they come and talk to us.
“Communication is the key.”
SNP councillor Gordon Clark, whose Rutherglen Central and North ward covers Burnhill, also believes the plans will be well-received.
He said: “I think in general it will be welcomed but obviously there are going to be parking issues and other issues relating to what happens on a match day.
“These will have to be closely examined so there isn’t undue disruption.
“This will be a major stadium, not just a junior ground, but I think if everything is smoothed out then it will be good for both clubs as well as the burgh.”
Clyde still have an option on the table to move to the Showpark in East Kilbride.
They have played at Cumbernauld’s Broadwood Stadium since 1994.
They left Shawfield in 1986 and spent the intervening years playing at Hamilton Accies and Partick Thistle grounds.