If we try and think of slimes as animals in nature, they must be composed of cells or at the very least one cell. In fact slimes do show an uncanny similarity to single cellular creatures, but there is a problem. There is a limit to the distance between the nucleus and the other edge of a single celled organism. While we could combat this by having multiple nuclei distributed all over the place and there are animals that have this strategy; syncytium, or slime molds.
The slime needs to be multi-cellular, we do have multi celled "slime molds" called dictyostelium. It has a slug like face and crawls around as a discrete blob, but even though it is a good match, it is not a good candidate for what we want. Natural slimes mold are either too small or to in-cohesive. These are as close to the slime we need as nature gets, after this we need to speculate.
We know our slime has a firm shape, so it needs to have some kind of support structure, we also know it eats and has eyes, so it will need organ systems. Based on what we know, the slime will need to be a discrete organism, no less complex than your average jellyfish. In fact, let's consider jellyfish.
Jellyfish are soft bodied sea critters that use their "arms" to catch and eat food. Their body is shaped like a bell to push water and propel themselves. Jellyfish have to tissue layers; The ectoderm, forming their 'skin' and the endoderm, forming their gut, represented below by green and blue respectively.
So for our jellyfish slime, we are thinking of a jellyfish with a ectoderm that is spherical rather than bell shaped. Comb Jellies are jellyfish that match this description fairly well. Most comb jelly species are roughly ball shaped as they float around through the sea. When stranded on land, they form small transparent blobs,
Rather akin to the slimes we are after, but the slimes we want are large, with an height ranging from a meter to a foot, though minecraft slimes can and do exceed human height. To retain shape for such a large body a fluid filled body cavity in the shape of a torus is needed, surrounded by muscle to keep it under pressure and combat gravity wanting to flatten it while on land. The resulting shape will be wide in the mid section, flat on the bottom and rounded on the top, which matches all of its depictions excluding minecraft for obvious reasons.
The inner walls of the body cavity will be lined with cartilage akin to ribs, which like a shark, will replace bones. All the organs, excluding the brain, will go in this body cavity. The brain will not be a single organ but instead work like an octopus, with the neural system being distributed across the body.
We now need a mechanism to propel the body upwards like we see in the games, kind of like a pole vaulter. Imagine multiple rods similar to a vaulting pole lining the body cavity, you would need at least four, though six or eight would be better. In pole vaulting, the pole is bent down to store elastic energy, which is then released, launched the vaulter into the air. A central band of very stronger muscle goes in the center of the slime (which is also the reason for the body cavity being torus shaped) which pulls the rods and when released, vaults the slime.
But, this only launches the slime upwards, to gain direction a change must be made. Instead of having one muscle we need many, one for each rod. failing to use one muscle results in movement.
Slimes are shown to possess a mouth but lack any means of getting food towards said mouth. However, slimes do possess another tool that could be even better, a slimy outer surface. Being slimy on the outside, food particles would just stick to the skin and move towards the mouth. The constant flow of mucus towards the mouth (which is then swallowed and excreted again to avoid waste) can be explained by having out slime be surrounded by small, rgoughly beating hairs called cilia, like our lungs do, but on the outisde.