The Dane is keen to provide competition for close-season signing Wilfried Bony up front after releasing Luke Moore and telling Leroy Lita that he is free to find a new club.
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"We want to add one more who can give us that in a certain part of the game or in certain games," Laudrup told reporters.
"And [Odemwingie] is a player with pace who knows the league."
The Nigerian has been frozen out of the first-team picture at the Hawthorns after attempting to force through a move to QPR in the January transfer window and repeatedly criticising the club on Twitter.
Odemwingie joined West Brom from Russian Premier League outfit Lokomotiv Moscow in August 2010.