Seed is produced by the flowers, but not by the bulbs. I imagine you're talking about offsets, or bulbils, which grow around the original bulb. You can take those off, and pot them up, might be about 2 years before they're large enough to flower.

I don't think so, Bamboo. Remember that crocus flowers appear on 'tubes', the ovary being at the bottom of the tube beneath soil level. The ovaries are just starting to swell with seeds and will appear on the stems at about soil level. I have collected some seeds from ours but most are not ripe yet.
Alan, it is a pity that you have already lifted the bulbs (corms) as the seed may not yet be ripe. Leave the seed pods attached to the corms and keep in a container in a place without much light, eg the garden shed, and let them burst open themslves. Select out any nice fat seeds and sow them.

Ah, didn't think of that because the question said 'beneath the soil', so I jumped to offsets...

From seed, they'll take a lot longer to produce flowers.

The seeds that I have just been collecting are from the autumn flowering C. pulchellus. Most of the seed came from first time-of-flowering seed grown corms that I sowed in June 2012, flowering autumn 2014! I will say that I had remarkable germination and they did flower more quickly that expected.