Update: FFmpeg recently added
afftdn which uses the noise threshold per-FFT-bin method described below, with various options for adapting / figuring out appropriate threshold values on the fly.
anlmdn (non-local means) is a technique that works well for video; I haven't tried the audio filter.
Either of these should be much better than highpass / lowpass, unless your only noise is a 60Hz hum or something. (Human speech can still sound ok in a pretty narrow bandpass, but there are much better ways to clean up a broadband noise background hiss.)
ffmpeg doesn't have any decent audio filters for noise-reduction built in. Audacity has a fairly effective NR filter, but it's designed to be used with 2-pass operation with a sample of just the noise, and then the input.
The comments at the top of https://github.com/audacity/audacity/blob/master/src/effects/NoiseReduction.cpp explain how it works. (basically: suppress every FFT bin that's below the threshold. So it only lets signals through when they're louder than the noise floor in that frequency band. It can do amazing things without causing problem. It's like a band-pass filter that adapts to the signal. Since the energy of the noise is spread over the whole spectrum, only letting through a few narrow bands of it will reduce the total noise energy a LOT.
See also Audio noise reduction: how does audacity compare to other options? for more details of how it works, and that thresholding FFT bins in one way or another is the basis of typical commercial noise-reduction filters, too.
Porting that filter to ffmpeg would be a bit awkward. Maybe implementing it as a filter with 2 inputs, instead of a 2-pass filter, would work best. Since it only needs a few seconds to get a noise profile, it's not like it has to read through the whole file. And you SHOULDN'T feed it the whole audio stream as a noise sample, anyway. It needs to see a sample of JUST noise to set thresholds for each FFT bin.
So yeah, a 2nd input, rather than 2pass, would make sense. But that makes it a lot less easy to use than most ffmpeg filters. You'd need a bunch of voodoo with stream split / time-range extract. And of course you need manual intervention, unless you have a noise sample in a separate file that will be appropriate for multiple input files. (one noise sample from the same mic / setup should be fine for all clips from that setup.)