(CNN) -- Officials in Alaska, one of three states yet to certify winners in the November 4 U.S. Senate races, say they hope to have nearly all ballots counted on Tuesday.
And officials in Minnesota, home of one of the other unresolved races, intend to rule Tuesday whether certain rejected absentee ballots should be considered in a recount scheduled to start Wednesday.
In the Alaska race between embattled Republican Sen. Ted Stevens and Democratic challenger Mark Begich, about 24,000 ballots remained to be counted on Tuesday, said the state's elections director, Gail Fenumiai.
However, it's possible a few straggling absentee votes might come in Wednesday in time to be added to the tallies.
The race drew national attention, especially after Stevens was convicted in October of filing false statements on Senate financial disclosure forms. In early returns in the days after the election, Stevens -- the Senate's longest serving Republican -- held a narrow lead over Begich, who is mayor of Anchorage.
But Begich took a slim lead last week as officials sorted some 90,000 additional votes -- nearly a third of all ballots cast in the state. Those votes included about 60,000 absentee ballots, 9,500 early votes and another 20,000 "questioned" or provisional ballots being checking for validity.
By Friday, when vote counting was stopped for the weekend, Begich had 47.37 percent of votes counted; Stevens had 47.02 percent. The two were separated by 1,022 votes out of more than 290,000 cast, according to the Alaska Division of Elections Web site.
Alaska allows up to 15 days, longer than any other state, after Election Day for absentee ballots to arrive and be counted if they were postmarked by Election Day and mailed from outside the United States. Absentee ballots mailed inside the United States are accepted up to 10 days after the election.
Election officials said that schedule was adopted in consideration of Alaska's sprawling geography, sparse population and sometimes spotty mail service in remote areas.
"Wednesday is the last day we will accept absentee ballots, but we really don't expect many to come in," Fenumiai said Monday.
In Minnesota, vote totals last week showed Republican Sen. Norm Coleman 206 votes ahead of his Democratic challenger, Al Franken.
On Tuesday, the secretary of state's canvassing board is scheduled to hear a request by Franken's campaign that certain already-rejected absentee ballots be counted during a statewide hand recount scheduled to start Wednesday.
Asked what the campaign plans to do if the board decides it will not count rejected ballots, Franken spokeswoman Colleen Murray said the campaign hasn't ruled out anything, including asking for a postponement of the recount.
Georgia is the other state with a Senate race yet to be resolved. Republican Sen. Saxby Chambliss will face Democrat Jim Martin in a December 2 runoff.
CNN's Chris Welch contributed to this story.