If Angus King wants to join the GOP, Sen. Thad Cochran has a seat on the Appropriations Committee at the ready.

“If he’d like to become a Republican, let’s talk,” the Mississippi Republican quipped this week.

Though the former Maine governor is widely expected to caucus with Democrats if he wins his Independent bid for Senate this year, King has a chance to wield disproportionate influence in what is sure to be a narrowly divided chamber, regardless of which side he lines up with on leadership votes. And the courtship for his vote could be furious, even after he picks sides.

Cochran, who would welcome a return to the majority on the powerful Appropriations Committee, said the fawning over King could rival that of former Sen. Jim Jeffords (Vt.), who switched allegiances in 2001, handing control of the Senate — and the accompanying gavels — to the Democrats.

“He’ll be the favorite date,” Cochran said. “Everybody will be inviting him to dinner and trying to get to know him.”

Both party leaders could make elaborate offers to King even though Republicans doubt he would join their cause, given King has already said he supports President Barack Obama for re-election.

In an interview with Roll Call, King sounded receptive to negotiating. He said he would expect to see better offers from Majority Leader Harry Reid (D-Nev.) and Minority Leader Mitch McConnell (R-Ky.) the closer the balance of power in the chamber.

Reid is no stranger to the kind of horse-trading that may be needed to lure King into the stable. As Minority Whip in 2001, he agreed to cede his own claim to the gavel of the Environment and Public Works Committee to Jeffords if he switched.

King, however, said he has purposefully avoided discussions with Reid, Democratic Senatorial Campaign Committee Chairwoman Patty Murray (Wash.) or any other Democrats who could lead to a backroom deal.

“I haven’t talked to any Senators. And nobody on my staff has talked to any Senators. For the very reason that I want to be able to do what I just did and say, zero,” King said. “Within four days of my announcement, the Republicans made an ad about a backroom deal. They just made it up.”

Jeffords became an Independent who caucused with Democrats and maintained a streak of unpredictability for the remainder of his Senate career. On a special education bill, for instance, he pushed to get additional money for Vermont.

The current Maine Senators, Republicans Susan Collins and Olympia Snowe, have long been willing to cross the aisle and provide needed votes to Reid — usually in the majority’s quest to get 60 votes to overcome procedural blockades. That was especially true in the last Congress, when Reid had a larger majority and only needed to pick off one or two Republicans to kill a filibuster.

Snowe and Collins don’t profess to provide swing votes as a favor to the Nevada Democrat but, like Jeffords did, to act in the best interest of the New Englanders they represent.

While the 50th or 51st vote holds less power on a daily basis than it would without the threat of perpetual filibusters, a potential swing vote like King could prove critical during budget season. While King is an unlikely vote of support for any GOP-led Senate’s attempts to repeal the 2010 health care law, he could be an important factor in votes on other issues that are important to the majority or the minority.

Retiring Democratic Sen. Ben Nelson, a moderate deal-maker from Nebraska, knew King when both were governors of their home states. He said King has a chance to do what seems impossible and bridge the gap.

“I think he’s in a position to try to find others who will share thoughts with him, but it is very difficult these days to get anything done from the middle,” Nelson said. “No matter how hard you try or who you work with — but if anybody can do it, I think he’s the one who could.”

But as for the possibility of choosing no side at all, the woman who would be his senior Senator said that could leave King with the least influence of all.

“It’s my experience that to be effective, you have to join a party, and you have to be part of a caucus even if you don’t vote a party line,” Collins said.

“He has been very unclear,” Collins said. “You can’t have committee assignments without belonging to a caucus, so I really can’t speak for his approach.”

But King still leaves open the possibility that he may not choose at all. He said that based on his interpretation of the Senate rules, he could get committee assignments without joining with either the Democrats or Republicans, but how those would be apportioned is an open question.

Joshua Miller contributed to this report.