Coleman defeated former Vice President Walter Mondale by 40,000 votes in Tuesday's midterm election after Mondale replaced Wellstone on the ballot.
"There are benefits for Minnesota if I'm sworn in before any others," Coleman said. "Even a day before. I'm going to talk to Dean about the transition. We'll work it out."
Barkley, who has run unsuccessfully for Congress as an Independence Party candidate, arrived in Washington Thursday to represent Minnesota in the lame-duck session of Congress that convenes next week.
"The governor said, 'Go get what you can for the state,'" Barkley told the Minneapolis Star Tribune. He said he had not decided whether he will caucus with the Republicans, who will have a majority, or the Democrats who lost their 1-vote advantage in Tuesday's election.
"Quiet frankly, I don't have any personal stake in which party controls the Senate," Barley said.
Senate Republican Leader Trent Lott of Mississippi and Democratic Leader Tom Daschle of South Dakota both briefly met with Barkley, who could serve two months until the 108th Congress takes office in January.
If he steps aside earlier, Coleman would be able to get a jump on seniority on three member of the new class of freshman senators and perhaps get preferred committee assignments.
Coleman said he would like to be assigned to the Senate Agriculture Committee.
"If it gives an advantage to the state and if it gives him a leg up, certainly I'll consider it," Barkley said. Ventura, who did not seek re-election, has not indicated how he feels about Coleman taking office early.