The choice will be made, probably early next year, by committee Republicans but the loser can force a vote by the entire Senate.
Lugar, a moderate Republican, will give up the chairmanship when the Democrats take control of the Senate next year.
Helms, leader of the Senate's GOP conservative bloc, differs with Lugar on some policy issues. Most recently he opposed the tougher sanctions against South Africa that Lugar shepherded through his committee.
Helms also has been a strong supporter of certain right-wing Americans to become ambassadors.
In a letter to Lugar, dated Nov. 7, Helms said he could not support Lugar's request to serve as the top Republican on the committee.
Helms said he would not have challenged Lugar if the Republicans had retained control of the Senate but said "the ranking minority post is a different matter."
In his reply, Lugar said he was "sorry" that Helms would not support him and added, "Good luck and may the best man win."
Lugar and Helms have served on the committee the same number of years but Helms is credited with seniority because he was elected to the Senate four years before Lugar.
Two years ago, fulfilling a campaign pledge, Helms chose chairmanship of the Senate Agriculture Committee to protect North Carolina's interests in tobacco and peanuts, leaving the chairmanship of the foreign relations committee to Lugar.