I've received conflicting messages on this point -- on the one hand, I've been told that "forming a natural home for algebraic $K$-theory" was one motivation for the development of motivic homotopy theory. On the other hand, I've been warned about the fact that algebraic $K$-theory isn't always $\mathbb A^1$-local. By "algebraic $K$-theory,", I mean the algebraic $K$-theory of perfect complexes of quasicoherent sheaves (I think -- let me know if I should mean something else).
I'm pretty sure that Thomason and Trobaugh show that algebraic $K$-theory
always(in the quasicompact, quasiseparated case) satisfies Nisnevich descent.
Under certain conditions, algebraic $K$-theory is $\mathbb A^1$-local and has some kind of compatibility with $\mathbb G_m$ which should make it $\mathbb P^1$-local. I think
Weibel(already Quillen) calls this "the fundamental theorem of algebraic $K$-theory".
So putting this together, let $S$ be a scheme, and let $SH(S)$ be the stable motivic ($\infty$-)category over $S$.
Is algebraic $K$-theory of smooth schemes over $S$ representable as an object of $SH(S)$?
How about if we put some conditions on $S$ -- say it's regular, noetherian, affine, smooth over an algebraically closed field? Heck, what if we specialize to $S = Spec(\mathbb C)$?
Does it make a difference if we redefine $SH(S)$ to be certain sheaves of spectra over the site of smooth schemes affine over $S$ or something like that?