An intense race is developing between Republican Sens. John Cornyn (Texas) and Norm Coleman (Minn.) over who will become the next chairman of the National Republican Senatorial Committee (NRSC).

Several Republican senators told The Hill that Cornyn and Coleman have asked for their votes in recent days.

“They both talked to me and they both asked me to commit,” said Sen. Mike Enzi (R-Wyo.), who said he would wait to make a decision.

This past week was the last chance for leadership hopefuls to solicit their colleagues face to face.

Senators are expected to adjourn at the end of this week and return to Washington on Nov. 17 for a lame-duck session. Votes on leadership posts would happen then.

Senate Republicans are expected to lose seats next month — some say as many as five, some say more.

The next cycle looks more promising for the party. While Senate Republicans will have to defend 19 seats in 2010 — four more than Democrats — most of those races take place in states that President Bush carried in 2004. Only incumbent Sens. Judd Gregg of New Hampshire and Arlen Specter of Pennsylvania, both battle-hardened Republican veterans, have contests in Democratic-leaning states.

The next chairman of the NRSC may also benefit if Sen. Barack Obama (D-Ill.) defeats Sen. John McCain (R-Ariz.) for the presidency. Historically, the party that loses the White House returns to have a strong showing in the next midterm cycle.

And as a bonus, the NRSC chief would have a chance to defeat Senate Majority Leader Harry Reid (D-Nev.), who is up for reelection and has seen low approval ratings in some polls.

Coleman’s vigorous behind-the-scenes maneuvering has raised eyebrows because he is in the midst of a close reelection race against Democratic challenger Al Franken.

A spokesman for Coleman said the senator is concentrating on his reelection campaign.

“Although many of the senator’s colleagues have approached him about a leadership position, he is solely focused on addressing the needs of Minnesotans, and reaching out to Minnesotans and asking for their support to represent them again for another term in the United States Senate,” said LeRoy Coleman.

Cornyn is also facing a competitive reelection bid against Democratic challenger Rick Noriega, a state legislator and Iraq war veteran, but that race is not as tight as Coleman’s.

Furthermore, Cornyn has bolstered his bid against Coleman by giving hundreds of thousands of dollars to the NRSC. He gave $250,000 of his campaign funds to the committee in late July. He claims to have raised $2 million this cycle and $5 million during his first term in the Senate.

Coleman has given $30,000 to the committee from his leadership political action committee (PAC). Officials have not asked him to give from his personal campaign account because of his tough race.

Coleman still has supporters from his unsuccessful run for NRSC chairman four years ago. He lost narrowly to Sen. Elizabeth Dole (R-N.C.), who won in part because of the influence of her husband, former Senate Majority Leader and GOP presidential candidate Bob Dole (R-Kan.).

Sen. Mel Martinez (R-Fla.), who spoke to both Cornyn and Coleman in “the last few days,” said he supported Coleman’s bid against Dole and remained committed to him.

The duel between Cornyn and Coleman is shaping up as the only Senate leadership race.

While House Minority Leader John Boehner (R-Ohio) may lose his job, several GOP senators and aides predicted that Senate Republican Leader Mitch McConnell (Ky.), Republican Whip Jon Kyl (Ariz.) and GOP conference Chairman Lamar Alexander (Tenn.) would keep their posts even if Republicans drop to 45 seats or lower.