There are two efforts that are particularly noteworthy!

The first one, which admittedly is surrounded in conspiracy theories and the like, involves Rudlof Hess. Hess was Deputy Fuhrer to Hitler from 1933, though did not have the same political ambitions as the rest of Hitler's inner circle. In 1941, he flew a plane to Scotland, in an attempt of negotiating peace with the Duke of Hamilton, who Hess believed was in opposition to the war.

Captured by the British, he was imprisoned for the rest of the war. Hitler was shattered by this betrayal, and stripped him of all titles and labelled him as a madman, though there are some writers who believe that it was a genuine attempt, in a prelude to the German invasion of the Soviet Union.

Heinrich Himmler, head of the SS, also attempted peace negotiations with the Western Allies in 1945, though by this stage the Allies were already within Germany, and total surrender was close by. Hitler once again did not approve of this, and flew into a massive rage, shooting the SS representative at the fuhrerbunker, who was also Eva Brauns brother in-law.

Sources: Heinrich Himmmler: A life, Peter Longerich

The Crown and the Swastika: Hitler, Hess and the Duke of Windsor, Peter Allen.

While I know that Peter Allen's book makes some grand claims about the affair, the factual information is correct, and also gives interesting examples of historical mystery solutions.

In addition to Mr. McCarthy's response, a number of the anti-Hitler generals tried to use contacts within the Allies to establish a peace treaty should Hitler be killed. The early ones were impossible for the Allies have fulfilled (a 1941 proposal by Carl Goerdeler, a plotter against Hitler and a monarchist politician, demanded pre-ww1 borders with France, Belgium, and Poland, and annexation of Austria and the Sudentenland). After that, the allies started demanding unconditional surrender from any possible peace deal. The plotters who got the furthest (the 20 July plot, commonly incorrectly known as Operation Valkyrie) still demanded in all of their proposals annexation of Poland as well as occupying the rest of Eastern Europe, and a continuation of the war against the Soviets.

There were two notable attempts by the German resistance within the Wehrmacht to prevent or to end the war.

The first one took place after Hitler made his war plans clear and forcibly retired 16 Generals who were opposed to his drive for war, and transferred 44 more. The head of the German General Staff Ludwig Beck tried to organize his peersbefore a meeting that took place on August 4th 1938. That meeting showed showed that with the exceptions of Generals Ernst Busch and Walter von Reichenau, the entire General Staff presciently considered a war to be unwinnable and eventually to lead into catastrophe. In particular, General Erwin von Witzleben made plans together with, among others, Generals Franz Halder, Walter Graf von Brockdorff-Ahlefeldt, Paul von Hase and Erich Hoepner to arrest, imprison and eventually put Hitler before a court in the event of a Franco-British declaration of war. At the same time, Admiral Wilhelm Canaris, who was working with Beck and von Witzleben, sent Ewald von Kleist-Schmenzin to Britain as his envoy, asking for a British declaration of war in the event of a Nazi invasion of Czechoslovakia. That declaration would have given the General Staff both the pretext and support for the above-mentioned overthrow of Hitler[1].

Chamberlain's caving-in at Munich however destroyed the German generals' plans. Beck was eventually pressured into retirement, von Witzleben was transferred to Heeresgruppe two and Canaris shifted his efforts to more clandestine means. All of them were and eventually killed after the failed 20 July plot.

The second attempt was initiated in 1943, when resistance within the Wehrmacht[2], through e.g. the British contacts of resistance member Wilhelm Canaris, tried to explore terms of peace if the resistance got rid of Hitler[3]. Churchill's reply, transmitted to Canaris, was simple: "Unconditional surrender". Likewise, Helmuth James Graf von Moltke tried to contact the British and get assistance for the German resistance, only to be rebuffed in the same fashion. Moltke also contacted Roosevelt through the OSS, reporting on the extend and plans of the German resistance. The result was the same as with the British: "Unconditional surrender".

This doctrine of "Unconditional Surrender" had actually been established by Roosevelt at the Conference of Casablanca in January 1943. For the German resistance, this doctrine was the most harmful obstacle in their recruitment efforts: They understood very well that it would mean the destruction of their country, even more loss of territory and people than after Versailles as well as the displacement and death of millions of their countrypeople. As long as it remained the Allies' demand they refused even to negotiate, German soldiers would fight to the end under whichever leadership; however if the Allies signalled they were ready to retract it in the event of a successful coup, a considerable number of German commanders would have joined the resistance and the war - at least in the west - would have been over.

[1] Terry Parssinen: The Oster Conspiracy of 1938: The Unknown Story of the Military Plot to Kill Hitler and Avert World War II. 2004, Harper Perennial - ISBN 0060955252

[2] Klemens von Klemperer: German Resistance Against Hitler: The Search for Allies Abroad, 1938-1945. 1994, Clarendon Press - ISBN 0198205511

[3] a) Richard Bassett: Hitler's Spy Chief: The Wilhelm Canaris Mystery. 2005, Orion - ISBN 0297846876. b) Heinz Hohne: Canaris: Hitler's Master Spy. 1979, Doubleday - ISBN 0385087772

Not a direct answer but a bit of important context and an inflection point which hasn't been brought up so far - the Casablanca Declaration at the beginning of 1943 essentially precluded any resolution of the Allies' conflict with Germany short of unconditional surrender. A short summary can be found here:

https://history.state.gov/milestones/1937-1945/casablanca

This policy was a cornerstone of allied unity although it does not seem to have done much to shake Hitler's and other Third Reich leaders' conviction, held almost to the very end, that the alliance is fragile and that splits within it will provide Germany an opportunity to avoid defeat.

Yes straight after the fall of france Hitler offered total cessation of the war in the West. Germany would evacuate all of France except Alsace and Lorraine, which would remain German. It would evacuate Holland and Belgium, retaining Luxembourg. It would evacuate Norway and Denmark. In short, Hitler offered to withdraw from Western Europe, except for the two French provinces and Luxembourg [Luxembourg was never a French province, but an independent state of ethnically German origin], in return for which Great Britain would agree to assume an attitude of benevolent neutrality towards Germany as it unfolded its plans in Eastern Europe. In addition, the Führer was ready to withdraw from Yugoslavia and Greece. German troops would be evacuated from the Mediterranean generally and Hitler would use his good offices to arrange a settlement of the Mediterranean conflict between Britain and Italy. No belligerent or neutral country would be entitled to demand reparations from any other country, he specified.

The proposal contained many other points, including plans for plebiscites and population exchanges where these might be necessitated by shifts in population that has resulted from the military action in Western Europe and the Balkans. But the versions circulating in authoritative circles all agree on the basic points outlined above.