Assuming the painting is at least six months cured after the last brush stroke was deemed dry to the touch:
1. Wipe the surface with a different soft cloth dampened slightly with distilled water. The distilled water wipe should remove all the water-soluble dirt. Fly specs and other dirt are water-soluble.
2. A day later, wipe with a soft cloth slightly moistened with odorless mineral spirits, which should remove any oily deposits (e.g., oil from hands)
3. Allow another day for the solvent to evaporate before applying varnish.
Incidentally, the advice on how to clean oil paintings prior to varnishing came from the late Ross Merrill, my friend and ASTM colleague who was Chief of Conservation at the National Gallery in Washington DC. On one painting I experienced the beading up of the varnish in places, so I had to remove all the varnish and start over. Following Ross’s advice in cleaning the surface, the varnish went on perfectly with no beading.
Clean a painting prior to varnishing. Sometimes low molecular weight varnishes will bead up if the surface of the painting isn’t clean. Regalrez varnishes like Gamvar and Rublev Conservar are low molecular weight varnishes.
Alcohol should not be used on an oil painting except by expert conservators.
Gamsol is the mildest kind of odorless mineral spirits, and used as I advised in my book, that is, by gently wiping the surface of the painting with a soft cloth very lightly moistened with it. There should be no adverse effects from this procedure if the painting has had at least six months to cure. This is not theory or guessing on my part, but is based on actual experience doing it at the suggestion of Ross Merrill when he was Chief of Conservation at the National Gallery in Washington DC.