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During the summer of 1940, Robert Lusser returned to Heinkel from Messerschmitt and immediately began work on a new high-speed bomber project designated P.1055. This was a shoulder-wing cantilever monoplane; the tailplane, which had considerable dihedral, had twin fins and rudders. The glazed canopy of the cockpit was faired into the nose of the aircraft and provided excellent external visibility for its two occupants, who were seated in an atypical back-to-back configuration. Many parts of the airframe, including its single-spar wing, rectangular-section fuselage and tail surfaces, were made of metal and had a stressed-skin covering.
The P.1055 was originally intended to be a multi-purpose aircraft, and was a relatively advanced design for the era, with a pressurized cockpit, twin ejection seats (the first to be planned for use in any combat aircraft), tricycle landing gear and remotely controlled, side-mounted FDSL 131 defensive gun turrets. Power was to be provided by a pair of DB 610 "power system" engines producing 2,910 hp each, delivering a performance with a top speed of approximately 750 km/h and a 4,000 km  range with a 2,000 kg bomb load.

In August 1940, an initial review by the Reichsluftfahrtministerium rejected the P.1055 considering it to be too complex and risky. Lusser quickly offered four versions of the fighter with various wingspans and engine choices in order to balance performance and risk.