Answer: Morning glories really don't like to have their roots disturbed so you'd be taking a chance if you try to move them. If you really value the plants, why not wait patiently to see if they flower and produce seeds later this summer. If so, you can collect the seeds and plant them near your fence. If not, you may be able to move the vines in the fall when temperatures are cool and the soil is quite moist. If you dig up a large area of soil you may not disturb the roots too much and the morning glory may not even know it's being moved.
If you're convinced that you'd like to try moving one, water the soil well the day before, and move the vine the following evening (after the hottest part of the day). After you've replanted it, water it well to eliminate any air pockets and help settle the soil. If you're successful in moving one vine, you can try to move the second vine. Best wishes with your garden!
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