The affricate /t͡s/ is not considered to be one of the basic sounds of English (it's not an English "phoneme"). English speakers do undeniably make use of a consonant cluster /ts/: you can find it at the end of plural nouns like bats or crates or third-person singular verbs like hits or fights. It just isn't considered to be a sound of its own: it's just a combination of the /t/ sound and the /s/ sound.
The consonant cluster /ts/ doesn't occur at the start of any native English word. It's not particularly difficult for an English speaker to pronounce /ts/ in this position, but I'd compare it to other consonant clusters that don't occur at the start of native English words, like /ps/ or /kʃ/.
Foreign words that are spelled with the letter Z may be pronounced by English speakers with /z/, even if the original language had another sound. This is the case for zucchini (pronounced /zuˈkini/ in American English) and influenza. In some words like zeitgeist, you might hear either /z/ or /ts/. Rarely, this kind of variation leads to confusion in the other direction: the word chorizo, where Z represents /θ/ or /s/ in Spanish, is apparently pronounced as "choritso" by some English speakers.