How to select objects by circling it without moving an object which is on the background?

If, in attempting to marquee-select an object, you click/drag on a different live, unlocked object, you're going to move that other object; it's practically a 'law of physics'. Instead, you could:

• Lock the background object (not sure why you refuse to consider it)
• Marquee-select, starting away from both objects, dragging across both; press shift and repeat, stopping short of the object you want to leave selected
• Marquee-select, starting away from both objects, dragging across both; press shift and click on the background object to deselect it and leave the target object selected

danezeq,

Do you mean NOT to circle the objects?

I did, but that was when you said object (meaning one object) which can be selected in full by Alt/OptionClicking with the Direct Selection Tool (sorry, I forgot to mention that), even if the object is part of a Group.

When you are saying objects, as in more than one, you can hold Shift from the beginning to select multiple objects by Alt/OptionClicking (or just Clicking with the Selection Tool if applicable).