We built a new bathroom in the basement last year. In that corner on the outside of the house, if the drain pipe gets clogged, water will end up in the basement where the new bathroom is. It happened once, and now there must be trapped water under the linoleum, and the cement under it has softened and bubbled up in a couple of places. It has been about 6 months since the water leaked into the basement, and I just noticed a new spot is starting to soften, so the water must still be there. Do we need to remove the linoleum and let it dry out? And maybe remove some of the cement floor where it is damaged?

Most likely the linoleum will need to be taken out, thrown away, any mold cleaned up and new flooring placed down. The cement floor should not have any damage to it, if it is damages, you will have to break up the areas where it is bad and repour those areas

If it hasn't dried up in 6 months you might have a ground water problem. You might look into installing a sump pump. Good luck, Tom