The meeting on Wednesday will involve Liverpool chief executive Rick Parry and Everton chairman Bill Kenwright and Caborn is likely to back the idea.
Caborn has influence with the Regional Development Agency who could provide some funds and Liverpool City Council are also keen to promote sharing.
"Mr Caborn has had a request to convene a meeting and he is more than willing to talk through the issues," said a spokesman for the Department of Culture, Media and Sport.
"He is generally behind the concept of ground-sharing and has supported the recent agreement between Leicester City and Leicester Tigers."