If it comes up the basement drain, then the obstruction is further down, and so you can’t really flush it from the drain and out. You need to find the sewer service well further out and flush it from there and in.

It’s likely a broken pipe taking in sand or branches, so flushing it won’t be a permanent fix. A plumber can probably fix it from outside too, so it doesn’t have to be very expensive.

You don’t need to waste your money on snakes and that kind of stuff. If a regular hose on full blast can’t loosen the obstruction from outside, then you need a plumber anyway.