The Mariners previously had a bid rejected for the pair, but have now come to an agreement after they handed in transfer requests.
Osborne, 24, has signed a two-and-a-half year deal while Asante, also 24, has agreed an 18-month contact.
They are reunited with boss Marcus Bignot, who took over in November.
Dutch forward Asante, who has 12 goals this season, and midfielder Osborne are both available to face Notts County on Saturday.
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Grimsby Town have signed midfielder Jamey Osborne and striker Akwasi Asante from National League side Solihull Moors for undisclosed fees.