The Daily Star says West Ham have joined the race to sign Koch.
Tottenham, Norwich and Leeds have all sent scouts to watch the 23-year-old German international in action this season.
But the Hammers have also now asked his representatives to be kept informed of any developments.
And Koch is the right age and profile as Moyes targets younger players with potential to grow.
Freiburg turned down £17.5m for Koch from Leipzig in January but agents believe he could now be available for less with prices tumbling due to coronavirus.