I believe it is possible to have a needle threader fitted to a modern sewing machine, and many newer models have this feature.
I myself sew on a 1926 electric Singer and have solved your problem as follows:
Insert one of the wire threaders you use for hand-sewing into the eye of the machine needle. It should have the point of the threader coming out at the side or front of the needle where you insert the thread.
(Old machines like mine thread sideways through the neede from left to right, newer ones from back to front, which is why I fear the foregoing sentence may be unclear.)
You may need to use a cheap pair of reading glasses of a stronger strenght to do this, or a magniflying lamp.
Then take the upper thread, which you have already threaded the machine with up to this last point (threading the needle) and thread a crewel needle or embroidery needle with the upper thread, then pass the needle through the wire threader, drawing a good length through with the crewel needle. Draw the thread out of the crewel needle and draw the needle threader slowly out of the machine needle. Keep a firm hold on the loose end of the thread while you withdraww the threader!
If you have difficulty getting the upper thread through the holder on the front or side of your machine above the sewing needle, you can use the crewel needle trick there too.