It's a little more complex than this. Wage suppression does occur but only at the very bottom strata of employment, specifically those producing use-values that are directly consumed within the country where the labour is performed. Employment in industries producing globalised/exported commodities tend not to see wage suppression and often sees an opposite effect as the higher concentration of highly-qualified labour attracts more investment. All this is to say that the overall effect doesn't tell the whole story, and different sections of the bourgeois may have differing reasons for supporting/opposing immigration.