I am not sure how old the vinyl tiles in your basement are but I would definitely lean toward leaving them, unless they are new and off gassing and/ or there is some sort of unbearable odor. Most vinyl tile becomes fairly inert quickly after installation and off gassing is not a major issue (unless you are very sensitive and the vinyl is particularly strong at off gassing.)
Vinyl tile glued to concrete is generally a nightmare to remove. It is extremely labor intensive in most cases.
Given your cold floor and general budget concerns I would agree with your installer and say leave the vinyl tile.
You can put down a true linoleum floor (like Marmoleum).
The true linoleum floors will come as either sheet or floating floor formats. Get the floating floor which will help with keeping any sort of moisture away from the tiles and make them last longer. It will also give you yet another layer (above the vinyl) between your feet and that cold concrete.
Depending on your ceiling height we will also sometimes put a layer of insulation board down below the floating floor (maybe a half an inch or an inch.) Ask your installer about options for an insulation board underlayment to truly make your floor warmer.
If it were my house, that is what I would do! Best of luck.
Enjoy your new floor and remember Marmoleum offers some amazingly beautiful and fun color choices. Be brave! Choose a color you love!
For more information:
Read "What is the best way to seal/encapsulate vinyl asbestos tile?" a Q&A answered by Joel Hirshberg.