I had exactly this question: that is, I needed a reference for this fact. Of course, doing internet searches now lead to this exact MathOverflow post! So I looked further and found a almost reasonable reference (copied from MathSciNet):

Solomon, Louis(1-WI) - An introduction to reductive monoids. Semigroups, formal languages and groups (York, 1993), 295–352. NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., 466 Kluwer Academic Publishers Group, Dordrecht, 1995 ISBN:0-7923-3540-6

On page 297, line -17, Solomon says that the Zariski closure of a monoid is again a monoid. However, it seems to me that the reference he gives (in Footnote 5) is broken (namely, reference [25] has no Lemma 2.1). On page 298, line 2, he says that the Zariski closure, intersected with $\mathrm{GL}(n)$, is a group. This time the references (in Footnote 6) look good to me.

I also had this question: that is, I needed a reference for this fact. Of course, doing an internet search now leads to this exact MathOverflow post! So I looked further and found an almost reasonable reference (copied from MathSciNet):

Solomon, Louis(1-WI) - An introduction to reductive monoids. Semigroups, formal languages and groups (York, 1993), 295–352. NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., 466 Kluwer Academic Publishers Group, Dordrecht, 1995 ISBN:0-7923-3540-6

On page 297, line -17, Solomon says that the Zariski closure of a monoid is again a monoid. However, it seems to me that the reference he gives (in Footnote 5) is broken (namely, reference [25] has no Lemma 2.1). On page 298, line 2, he says that the Zariski closure, intersected with $\mathrm{GL}(n)$, is a group. This time the references (in Footnote 6) look good to me.