Senate Minority Leader Mitch McConnell joked recently that Republican gains on Nov. 2 have finally put him on President Barack Obama’s “speed dial.”

But even though the president called the Kentucky Republican twice during the week of the election, their relationship still has a ways to go. And it’s likely to be tested as soon as Congress returns to work for the lame-duck session to deal with divisive issues such as taxes and spending.

“The lack of the most elemental relationship between the president and the Minority Leader plays out day after day after day,” Senate GOP Conference Chairman Lamar Alexander (Tenn.) said in a telephone interview last week.

Congressional Republicans have been scratching their heads for nearly two years on why Obama has not reached out to them more, particularly in the Senate, where the minority can stall crucial legislation.

“There’s a price to pay when you don’t develop that relationship. They need to know each other, and that takes getting together more than once every two years,” Alexander said.

Prior to the phone calls on the night of the elections and on Nov. 5, McConnell had met with Obama as president just once one-on-one — on Aug. 4, a year and a half after Obama took office. Before that meeting, Obama and McConnell had only spoken at larger White House meetings attended by Democrats. Previously, the pair’s last one-on-one conversation had occurred before Obama was inaugurated.

Neither man has sent public signals of wanting to build a closer bond. McConnell has repeatedly said he wants to make Obama a one-term president, while Obama called Republicans “enemies” shortly before the elections. Both men have backtracked from those comments, however.

In a Nov. 5 appearance on CNN, McConnell clarified, “It’s not particularly noteworthy that I would like to elect a Republican president in ’12, and he would like to have a second term in ’12. What is more important to the American people is what are we going to do between now and then. I think we can do some business.”

But since securing six more seats in the Senate and increasing his minority from 41 to 47 Senators, McConnell has made it clear he expects Obama and Congressional Democrats to “move in our direction,” which suggests he expects more give from the president than he does from his own ranks on legislative agreements.

White House spokesman Bill Burton said the president is optimistic about building a stronger relationship with McConnell despite the partisan nature of McConnell’s recent comments.

“The American people want both parties to work together,” Burton said. “Hopefully, that doesn’t get mired in Sen. McConnell’s belief that his No. 1 priority is to stop the president from getting re-elected.”

Burton added that Obama plans to continue inviting McConnell to the White House for both one-on-one and joint meetings with other Congressional leaders.

Regardless, a McConnell source said the Minority Leader took his Aug. 4 White House meeting seriously, asking the president to refrain from leaking the subjects they might discuss. That agreement was largely adhered to, except for an apparent White House leak in October to the New York Times, which reported the session was about judicial nominations and lasted more than 30 minutes before being interrupted by one of the president’s daughters.

The Times also insinuated that McConnell got that one-on-one meeting only after former Senate Majority Leader Trent Lott (R-Miss.) relayed the need for such a get-together to his friend, former Senate Majority Leader Tom Daschle (D-S.D.), who is also a presidential confidant.

The McConnell source would not confirm that the meeting, held just days before the Senate voted to confirm Supreme Court Justice Elena Kagan, centered on judicial nominations or any other details. 

Regardless, McConnell appeared to be testing Obama’s trustworthiness in asking that staff not reveal the topics discussed.

There are a few glimmers of hope for the Obama-McConnell relationship. The president’s Nov. 5 call was not about the elections but about a trade agreement Obama was hoping to strike with South Korea on his trip to Asia. Though talks on the trade deal broke down last week, the McConnell source said the Kentuckian had been prepared to issue a press release supporting the president’s efforts.

But Alexander said Obama missed opportunities to strike deals with McConnell over the past two years, including on a stronger debt commission proposal this year.

Though McConnell decided to oppose it, Alexander suggested that McConnell might have been amenable to sharing with the president members of his party who were generally supportive of the proposal. In the end, several GOP Senators who had backed the commission voted against it, causing the amendment to fail.

Alexander noted that successful presidents have seen wisdom in reaching out earlier to Minority Leaders. President Lyndon B. Johnson, he noted, called Senate Minority Leader Everett Dirksen (R-Ill.) every day.

Alexander blamed the lack of a relationship between the president and McConnell on the Democrats’ overconfidence and the large Congressional majorities that followed Obama into office in 2008. Democrats “were so consumed by their big majorities … that they ignored Republican points of view,” Alexander said.

Other Republicans said they are not sure where compromise will come from, given the White House has focused its GOP outreach on a handful of centrists and a few other rank-and-file Republican Senators.

Sen. Orrin Hatch (R-Utah) said in a phone interview last week that the president “talked a lot … about wanting to work with Republicans, but he didn’t offer any glimpse of how or where to do that.”

For example, Hatch blamed the White House for abandoning a bipartisan deal on the children’s health insurance law. He and Senate Finance ranking member Chuck Grassley (R-Iowa) had bucked GOP leaders and President George W. Bush by backing the deal in 2008, only to see the compromise get scrapped under a Democratic president the next year.

“We acted in total good faith … and the first thing they did was totally ignore us,” Hatch said.

Still, Hatch said that if Obama “turns to the center, then there’s some hope.”

McConnell has said he sees openings for bipartisanship on various trade deals, nuclear power, funding “clean coal” technology, on taxes and spending, and on reauthorizing Bush’s signature No Child Left Behind Act on education. Obama has indicated a willingness to compromise during the lame duck on extending Bush-era tax cuts, which McConnell has praised, and Democrats said they hope Republicans will work with them to create construction jobs during the reauthorization of the highway bill next year.

But no future private meetings between the two have been scheduled. However, McConnell is expected to attend the bicameral, bipartisan leadership meeting at the White House sometime next week.