Bolsonaro said his son, who now serves as a federal representative in the Chamber of Deputies, or the House, is qualified for the position.
Eduardo met with Foreign Minister Ernesto Araujo on Friday, who he said "expressed his support" for the appointment.
Eduardo would want to resign as a congressman if he did soak up the set of ambassador.
At the same meeting, the president said that in his candidature for re-election at the United Nations Human Rights Council, Brazil will adopt an agenda based on the "exclusion of gender mentions".
Eduardo, the third of the president's four sons, said he had yet to receive a formal offer but would accept the role. His appointment would need to be approved by the Senate Foreign Relations Committee before passing to the full upper house for confirmation.
The Brazilian embassy in Washington has been left without an ambassador since April when Bolsonaro removed the previous one.
Carlos Bolsonaro, another son of the president and a Rio de Janeiro city councilman, has taken a role in his father's social media communications and stirred controversy by attacking members of the Brazilian Cabinet.
Bolsonaro, who is following in his father's ideological footsteps and is an informal foreign policy advisor to the president, would have to resign from the Chamber of Deputies if he accepts the position.
Eduardo, now an MP, accompanied his father to a private meeting with Trump during a diplomatic visit to Washington in March.