Eli married his college sweetheart, Abby McGrew, on April 19, 2008. They don't have any children.
Source: http://www.nydailynews.com/gossip/2008/04/19/2008-04-19_giants_eli_manning_married_in_mexico.html
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Eli Manning and his wife have three daughters:
He and his wife, Abby, have announced that they are going to have a baby, so yes. It seems as if he does want children and if he didn't, then he's getting one