Well, no, Nasm isn't going to do any "conversion" for you. In theory, the C code should compile and run on Linux. In practice, it probably isn't 100% "standard" C, and probably contains "extensions" that won't work on Linux.
However, we can almost certainly write a program in Nasmese that will do the same thing as the 16-bit dos C code. If it's a fairly small program, this should be easy. If it's a large program... "tedious", at least. Depends... graphics, in particular, is a PITA in Linux...
What's this code do, anyway?
Best,
Frank