Yes, you can do that, but not with a segue, since they always instantiate new controllers (except for the unwind). So, you can create a property for your game view controller in the main menu, and only instantiate it if it doesn't already exist, then push it with pushViewController:animated:. Something like this,
-(IBAction)gotToGameView:(id)sender {
if (! self.game) {
self.game = [self.storyboard instantiateViewControllerWithIdentifier:@"Game"];
}
[self.navigationController pushViewController:game animated:YES];
}