Under the changes, the four provincial winners would be joined in the two groups by four qualifiers.
This would mean the provincial winners having to play three group games as opposed to just one in order to secure an All-Ireland semi-final place.
Kerry have made the All-Ireland semis this year after beating only two teams.
The Kingdom defeated Clare and Tipperary in the Munster Championship before meeting Clare again in the All-Ireland quarter-finals last weekend.
A GAA statement said the proposed changes would "provide a more exacting pathway to the All-Ireland final".
"This will have the effect of ensuring that the finalists will have been equally tested and that the two best teams in the country contest the All-Ireland final," said the GAA.
"Both All-Ireland semi-finals would be played over the one weekend which should generate greater excitement and also ensure both teams have the same period of time to prepare for the All-Ireland final."
The proposals also include a provision for Division Three and Four teams being granted home advantage in games against Division One or Two sides in the opening three rounds of the qualifiers.
The round-robin groups would see each county having one home game and one away match plus a game at Croke Park and an overall increase of eight matches in the championship.
If teams are level on points at the conclusion of the two groups, the result of the game between the sides would be used to separate them or in the event of a draw, the higher scorer in the table would be given the verdict.
If the changes are adopted, extra-time will be played in the event of a draw in all football championship games while the National Football League semi-finals will be abolished.
The current championship structure has been in place since 2001.
The GAA have unveiled proposed Football Championship changes which include the replacement of the All-Ireland quarter-finals with two round-robin groups.