Question regarding preferences in Gale and Shapley (1962)

Is it correct to say that preferences in the classic Gale and Shapley College Admissions problem are quasi-linear?

Or is this something thats introduced later in the literature, vis a vis Shapley and Shubik (1971), Kelso and Crawford (1982), etc?

Thanks

In Shapley and Shubik (1971) monetary transfers enter linearly. However, I wouldn't call this quasi-linear utility, because in quasi-linear utility most people have the following functional form over a good $$x$$ and money $$m$$ in mind: $$u(x,m) = v(x) - m$$, where $$v$$ is a continuous function. In their model, the $$v(x)$$ is simply the money value of house $$x$$, where the houses are district options.