In French, when spoken on its own, cent (“one hundred”) is pronounced /sɑ̃/, and euros is /øʁo/. When the two are combined, and although I’ve heard /sɑ̃øʁo/ a number of times, a linking consonant is often added between /ɑ̃/and /ø/. Sometimes a simple glottal stop (optionally preceded by a short pause before the second word) will do the trick: /sɑ̃ʔøʁo/. Glottal stops have no linguistic status in French, so, to the native speaker, it doesn’t quite feel like adding a consonant.
Another strategy is to insert a liaison which, more often than not, comes down to ‘sounding’ the final letter of the first word. Knowledge of the spelling is indispensible here: the ‘correct’ liaison would be /sɑ̃tøʁo/. Again, this liaison is optional, but there’s a tendency for liaisons to be more frequent in the speech of the higher classes.
I’ve very often heard /sɑ̃zøʁo/; and this is not restricted to euros; I’ve heard the spurious /z/ after cent before other words starting with a vowel (e.g. affiches). However, I’ve always heard cent ans - either in Guerre de Cent Ans (“Hundred Years War”) or elle a cent ans (“she’s a hundred years old”) - with a linking consonant, and the ‘correct’ one too: /sɑ̃tɑ̃/.
So why is it that we so often hear /sɑ̃zøʁo/? The most likely answer is (call it what you like) frequency/usage. The phonetic sequence /sɑ̃z/ is very frequent, both through the use of sans /sɑ̃/ (“without”) followed by a vowel (sans âme, /sɑ̃zam/, “without soul”); and the fact that when there is more than one hundred, cent is in the plural: deux-cents euros /døsɑ̃zøʁo/ (“two hundred euros”). So, to put it simply, /sɑ̃z/ is probably stored in our minds as a whole sequence due to its frequency. And for the same reason, it is probably available as a default chunk when we speak. Interestingly, vingt (“twenty”) is not subject to the same phenomenon. At least, in my own experience, people would either produce the correct /t/ in the singular /vɛ̃tøʁo/ or not produce any linking consonant; and in the plural, quatre-vingts euros is correctly produced with a linking /z/ or no linking consonant at all.
I’ve come across an additional type of liaison that is produced by old speakers in southern French: /sɑ̃nøʁo/. The actual spread of this variant has yet to be precisely defined; I feel that very few people use it. Also, an accurate phonetic and phonological description is, AFAIK, still lacking. The phonological form I’ve used here, /sɑ̃nøʁo/, is an oversimplification: in southern French, nasal vowels are perhaps more accurately described as a succession of oral vowel + velar nasal. So, my impression is that cent euros here is actually [saŋnøʁo]; or maybe [sanøʁo]? Anyway, I find this intringuing and I’ve only noted this recently although I’ve known the speakers in question forever.