Now, during the next rain storm, don your rain hat, boots, and rain coat, and walk around the ENTIRE perimiter of the house looking UP as well as down to see where water is running. You may find water running over, or behind, your gutters. You may find leaks at the seams. You may find a roof line that's MISSING a gutter. You may find a downspout is disconnected and leaking. You may find that the water running out the downspout is actually pooling at the foundation or running away, and eventually back TOWARDS the foundation. The 'skinny' is - Fix these problems and you'll solve about 80-90% of your moisture problem.
Gutters may need to be adjusted or rehung or even replaced in order to 'catch' water running off the roof. They may be clogged up w/leaves and in need of a cleaning. If you replace any gutters, get seamless gutters (with hidden hangers.) If water is pooling at the base of the downspout, buy at LEAST a 3' downspout extension (at the Home Center.) The further you can run the downspout, the better. (The recommended extension distance is actually 10'.) If the landscaping around your foundation is sloping TOWARDS the house, that's bad news. You want the opposite effect. Prune back your bushes and plants from the foundation too, at least 2'. This will allow air and sun ALL around the perimeter to dry things out quickly (once it stops raining.) French Drains are an option too. ALL this is a LOT cheaper than installing a sump pump or pouring new concrete. You just need to wait for the next heavy rain. IF a neighbor's water is running onto you lot, that's another problem that I'll skip for now.
RE: Underground spring - I would doubt it. If this was the case, you'd have STANDING water in your basement. I'd say more if you had a sump pump and it ran 24 hrs. a day, 7 days a week. 90% of the causes of water in the basement are due to the 'things' I've mentioned. Very, VERY rarely is it due to an underground spring. If you have more ?'s, post back.