Benjamin Moore has an 'Advance' water-based 'alkyd linseed oil on the market.
It is $28( USD)/ gty. and found in 'Ace Hardware' stores in the USA market.
I bought some 'white satin' to try out as a binder. Sure, it is great with commercial oil paint additions, watercolors and gouache. I use the paint formulations in studies on poster board. When the water evaporates, you have an oil paint. I find it dry to the touch in about 30-40 minutes, much slower than normal acrylics. Benjamin Moroe says , in home decore, allow 10-14 days for full drying. No Turpentine/ mineral spirits involved. Acrylics are not 'safe' for health or in the environment and too much is dumped out there.
I found inexpensive 'food coloring' dissolves readily and you can see the blue hue which is the food coloring drying in the linseed oil! A very inexpensive oil paints for studies and artists- young and old- on a budget.
I will also be exposing dried swatches to UV radiation to gauge fugitive properties of the dried food coloring paint. We are also not creating masterpieces here!