You can't exactly transplant the mushroom, but you could move some of the fungus that produced it to a new location where it might take hold. The mushrooms are the fruiting body of the fungus, producing spores similiar to how a fruit produces seeds. The main body of the fungus, the mycelium, is the white threads of fungus sprawling underground. You could try digging up some soil around the mushroom, looking for clumps of white threads running through the soil, and moving those to a new area. Once moved the mycelium will probably have to re-establish itself before it will fruit again and make more mushrooms. This can be tricky since you might pick up other fungus when you move the soil and the one you want would have to out-compete any that come along with it or any already in the area. This is why when you buy fungal inoculant for cultivating mushrooms it's usually grown up first as a pure culture in a relatively sterile setting, and you use an excessive amount of it compared to the amount of substrate. But, it is possible it could work.
That is all assuming the mushrooms you want to move from a fungus that gets it's nutrition from decomposition in the soil. If it's a mycorrihizal or parasitic fungus getting nutrition directly from the plant roots it will be very difficult to relocate. Which is why some cultivated fungus like truffles are so deadly expensive, they're extremely hard to cultivate.
Once upon a time I studied mycology pretty extensively :). I love the subject but I'm rusty on my identifications and just started trying to grow my own for food.
In most mushrooms the part that produces the spores is underneath the cap. If you look on the underside you'll see either gills or little pores, this is where the spores are produced. If the cap is fully opened it probably has spores. Other mushrooms produce the spores in different places, like with morel mushrooms they're in the dark little hollows all over the cap, in puffballs they're in the center and puff out the top when mature, etc. The spores are tiny though, and will probably have an even harder time competing with other microbes in the area than the mycelium.
If you want to collect some anyway you can make a spore print, which is a good way to help identify the mushroom. Cut off the stem flush with the cap and place the cap on a piece of white paper, then put a bowl or glass or something on top of it to keep out air currents. Let sit for a few hours and if the mushroom cap is mature it will shed the spores onto your paper in the same pattern of the gills. The color is used in identification.
That one is gorgeous! It looks like it an Amanita caesarea, or Ceasar's mushroom. If it is, it'll have a white spore print. Also it's mycorrihizal, so you wouldn't be able to transplant it, but hopefully you'll get more in the area. I'd be thrilled if I found one of those, beautiful!
Thank you. I think they are beautiful too :) There are many in the area they are in, just wanted to move them closer to the house. I have seen the big puff mushrooms here also. Do you use black construction paper for the spore prints? Thank you.