I would think so unless the oil and water emulsify, which I'm guessing is what happens over a few days? When I put rags in water for a few days, the water is definitely oily so seems to be what is called emulsion.
So maybe at that point there isn't enough oil to oxidize and/or the oil has already been oxidized (which is the exothermic reaction that can cause combustion in a pile of rags. There is oxygen in water.). Or not enough oil to worry about the temp increasing. It sounds like the big problem is enough surface area to oxidize while also a pile tight enough that the heat can't escape -> more heat = faster reaction -> continue this trend until you reach the combustion temperature of the oil. Furthermore, it sounds like Linseed Oil oxidizes really quickly and has a low combustion temp which is why it is especially likely to combust while other oils are less so.
https://www.heads-up.biz/linseed-oil