There were a number of plans proposed for what was called the "Amerika bomber." These included everything from the expected to the wildly exotic. The expected designs included large four engined bombers of fairly conventional particulars.

On the other hand, the exotic included many strange ideas. There was a proposal for a bomber to carry a Do-217 as a "piggyback" operation. The host bomber would carry the Dornier as far as it could, then the second bomber would launch a strike on its target. After it was done, the crew would ditch in the Atlantic ocean and be picked up by a waiting submarine. There were also flying wing designs--the Horten H.XVIII and the Arado E.555. These planes would have looked very similar to the current B-2 Stealth Bomber.

However, none of these planes or plans were operational. With interwar theorist Walther Wever dying in 1936 and the Germans moving away from strategic bombing after their experiences in the Spanish Civil War, the Luftwaffe concentrated on tactical air power over strategic. Once WWII was underway, this focus continued. By the time the US entered the war, the need for fighters to protect the German heartland was so intense that such an expensive project as a long-range heavy bomber that it was pushed aside. None of the plans for such an aircraft got off the ground, so to speak.

The ME-264 "Amerika" was one bomber that was developed for the Luftwaffe primarily as a long range reconnaissance aircraft for U-Boat fleets but was later developed into a long range bomber prototype for the "Amerika Bomber" initiative which had the Luftwaffe calling for aircraft designs capable of attacking the Eastern U.S. Seaboard (in theory) if launched from air bases in Western France or the Azores.

Only three were built, but the design and production were scrapped as Germany found it was in far greater need of fighter-interceptors to help combat the Allied Strategic Bombing campaign it had been undergoing, and Germany did not have the industrial resources to devote to both fighter and long-range bomber programs.

Source: Messerschmitt Me 264 Amerikabomber : the Luftwaffe's lost strategic bomber

The Wiki entry for the ME-264 also has a decent technical entry as well.

One plane that did go into service was the Heinkel He-177 .This however suffered from severe problems with the engines, which were paired in a single nacelle driving a single propeller on each wing (IOW it looked like a twin engined plane but in fact it had four). This led to overheating and fires- and the crazy insistence that an aircraft that size be capable of dive-bombing also led to delays in production.

By the time the problems were ironed out there was little time left before shortage of fuel grounded most of the Luftwaffe, though He-177s were quite successful bombing London in early 1944 and in Russia.