I have a JD 4105, which is pretty close to the specs for the 4044M,,
I also have the JD 390,, I use it on a different tractor,, but, for grins,, one day I put it on the 4105,,
Well,, that is a pretty useless combination,, I had to have over 300 pounds of front end weights to even drive the tractor at all.
I could not find the weight of the 370 or the 390 on the Deere website,, but the specs did show the housing on both are the same thickness,,
In my opinion,, I think the JD 370 mower should be used on a slightly larger tractor than the 4044M,,,