*I offered him with a raise is not correct. I don't know where you read that construction, but I don't recall ever seeing it.
Perhaps this happens sometimes when people confuse it with constructions like I presented him with something and I gifted him with something.
From the Oxford English Dictionary on the verb present:
11.a. trans. "To bring or place (a thing) before or into the presence of a person, or to put (it) into his hands, for acceptance; to offer, proffer, deliver, hand over, bestow, give (usually in a formal or ceremonious manner)."
13.a. to present a person with a thing = "to present a thing to a person" (sense 11). Formerly in the full extent of sense 11; now always implying bestowal of something as a gift to be kept (11 (a)). Also fig. "to furnish or supply with something". ...
... 1831 Southey Lett. (1856) IV. 247 Mrs. Bray...has desired to present you with a copy of Mary Colling's poem.
The verb offer has no such construction.