Is it possible to show pushViewController animation look like presentModalViewController but that has background functionality of pushViewCon
You should consider doing something like:
ViewController *myVC = [[ViewController alloc] initWithFrame:OFF_SCREEN];
[navigationController pushViewController:myVC animated:NO];
where OFF_SCREEN is some CGRect below the screen, like (0, 481, 320, 480). Then use an animation block to adjust the viewcontroller's view's frame.origin to be (0, 0). You would probably want to do the animate block in viewDidLoad or viewDidAppear.