The high court will consider the filing at a June 1 hearing, The Hill reported. The papers were in response to a brief filed by Democrat Al Franken's attorneys Monday, the newspaper reported.
In his reply, Coleman argued that a different set of standards were applied to absentee ballots across the state, which violated the equal protection and due process clauses of the constitution.
Coleman wants the court to order 4,400 rejected absentee ballots to be included in the vote count, which at last count favors Franken by 312 votes.
The newspaper noted that even if all of the 4,400 absentee were counted it would not guarantee a win for Coleman since some of the ballots would go to Franken.
"If you look at the precincts from which those votes come, which is about as reliable (a measure) as you can get, then they come overwhelmingly from precincts that Norm Coleman carried," Coleman attorney Ben Ginsberg was quoted as saying.