Reagan bargained with Gorbachev, but the two men never had a "walk in the woods." Achievement.org explains that the term involved another American and another Soviet:
After taking office in 1981, incoming President Ronald Reagan appointed [Paul] Nitze to lead the U.S. delegation to the Intermediate Range Nuclear Forces (INF) talks in Geneva, but for almost a year, Nitze and his team found it impossible to reach an agreement with the Soviet delegation. Finally, Nitze invited his Soviet counterpart Yuli Kvitsinsky to meet with him privately and work out a proposal to bypass the prior conditions both sides had imposed on the negotiating process. On July 16, 1982, the two men took their discussion to a wooded slope of the Jura Mountains, on the border between Switzerland and France. Away from the rest of their negotiating teams, the two diplomats crafted a fresh proposal to reduce the superpowers' stockpile of intermediate-range missiles. Nitze's bold act of personal diplomacy captured the imagination of the world press and inspired a long-running Broadway play, A Walk in the Woods.The incident happened three years before Gorbachev became General Secretary.