Hard to say without more pics or a schematic diagram. Based on this pic, neither location looks big enough to me.

Here is more photos of the yard.And the arbor we are looking at but we are painting it white. Arbor is roughly 7 foot wide and high by 3 foot wide. Each fence post is 8 foot apart on our fence. The arbor is 7 foot in length so it should fit inbetween the fence post but we need to bring the bed out another 3 foot if we put it where the fountain is now.. Just not sure there or down the fence more.
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If you desire a bit more privacy, setting the arbor at an angle facing the covered porch, with the end of the arbor set where the privacy fence ends would help and also create a more zoned space for the inner garden. You could bring the border out in a curve to accomodate it.

BTW: If you want that tree to become a lovely canopy, prune out those lower branches ASAP, before they get much larger. You want the lowest lateral branches at least 10 - 12 ft. above the ground on any shade tree and the younger and smaller it is, the easier it is to prune it to shape.

Is this arbor decorative or functional? If functional, I think location depends on your personal preference and the time of day when it will be most used. What do you prefer to see when sitting in the swing? Do you want to see the street and sidewalk so you can wave at the neighbors as the stroll by or do you want privacy? Also, if you use it during the afternoon, do you want to face the setting sun or way from it? We have an arbor swing which faces west. We us it mainly in the late afternoon and early evening. But is is covered by wisteria and partially shaded by a large oak tree. We are on a corner lot so we see the neighbors as they walk by on two sides. If it is only decorative, place it where you and the neighbors can most readily see and enjoy it.

We did cut some of those lower branches on the Maple last fall so the lowest branches on the left and on the right are now gone. We did not cut the second to last branch on the right off yet as that branch is thicker and we wanted the tree to get taller before we cut that one off..I like the idea of a higher tree.
As for the arbor I have not ruled out putting it on a angle like birdwidow said but I am still thinking along the fence looking at the knockouts roses and trampoline might be more my idea as I love to see my whole yard while swinging. It would need to come out a few feet though from the fence..I was thinking of putting a hosta bed under the Maple in the near future but not sure much would grow under there. Someone sent me a photo once of a flower bed under that maple and it looked kinda neat but wonder if the plants would really do that well there. Ill see if I can find that photo as it was kind of different..
Here it is
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