PARIS -- An American effort to persuade Britain's Lord Carrington to extend his tour as NATO secretary general has failed. The alliance now faces a divisive conflict in choosing its new civilian leader.

NATO needs a new note of discord as much as the Redskins need another injured running back. Protracted intra-alliance bargaining over the proposed U.S.-Soviet Euromissiles treaty has left deep scars on Chancellor Helmut Kohl's coalition government in Bonn, and new skirmishing could reopen these wounds.

But Washington's intermediaries could not persuade Carrington to put up with another year of commuting between London and NATO headquarters in Brussels as a way out of an unwanted public contest over naming his successor.

Margaret Thatcher's former foreign secretary, who has given notice that he will formally leave the NATO job next June, no longer masks his boredom with Brussels and his frustration with NATO's bureaucracy.

The opening round of this succession struggle pits Kohl's nominee, Defense Minister Manfred Woerner, against Norway's former prime minister and Conservative Party leader, Kaare Willoch.

It is a match between defense expert Woerner and accomplished political operator Willoch. But in a larger sense the struggle is along a dangerous fault line within NATO, with the alliance's largest nations compelled to back Kohl's choice while its smaller members are far more comfortable with the Norwegian. They also feel that it is again the turn of a small NATO country to have the secretary general post.

Woerner has shown a mastery of defense issues in his four years in Kohl's Cabinet. His handling of the deployment of Pershing II and cruise missiles in 1983 impressed American officials, as does his easy communication with the U.S.-dominated military command of NATO.

But Woerner has been on the losing side of the Euromissile debate since the deployment victory of 1983. He was deeply skeptical about Mikhail Gorbachev's "double zero" proposal this spring, and he fought a rear-guard action against Kohl's decision last month to phase out the aging Pershing IA missile launchers under West German control.

Kohl seeks to ease Woerner's anger and humiliation over these decisions, and to calm his restive right wing, by winning the NATO job for Woerner. Washington and Paris both will back him, fearing that another political embarrassment for Kohl on alliance affairs could be damaging in Bonn.

Britain's position is less clear at this point. Thatcher, who has never hit it off with Kohl, does not feel that she owes the chancellor any favors, allied diplomats say. Britain also may want to support the principle of rotating the job between large and small nations.

Norway's unusual announcement of Willoch's candidacy was apparently calculated to keep Britain from committing its support to Woerner. If Oslo can gain Thatcher's backing or neutrality, it can stop Woerner.

That, unfortunately, does not mean that Willoch automatically wins, even though he is the right candidate at the right time. His political skills are what the alliance needs at a time when it will be wrestling with Gorbachev's adept diplomacy, the aftermath of the Euromissile controversies and growing problems with Spain, Greece and Turkey on the southern flank.

Woerner's strong partisan involvement in the Euromissile debates would serve to keep contentious issues center stage in the badly needed debate on alliance strategy that a Soviet-U.S. treaty will bring. Moreover, his candidacy is undercut by the tepid support it is getting in official contacts from the West German Foreign Ministry, run by Woerner's longtime foe Hans-Dietrich Genscher.

Unable to agree on either Woerner or Willoch later this month when the special representatives gather in Brussels to review this question, NATO may temporize and then move to a "compromise" candidate like Belgium's foreign minister, Leo Tindemans, who has positioned himself for this scenario by dropping his active candidacy.

This would be like ordering rose wine to balance fish and meat courses at dinner. The compromise would please no one. And if drawn out, it would end up damaging Kohl and leaving Willoch's talents untapped by NATO. (Woerner would undoubtedly continue as defense minister in Bonn.)

Washington should throw its weight behind resolving this contest quickly and decisively, and, in the best of all worlds, with the early selection of Willoch. With Gorbachev's autumn peace offensive taking shape, NATO should not dawdle.