Attorney general Mohammad Is'haq Alko told reporters that Eikenberry had violated "diplomatic ethics" by ordering him to have the banker arrested.
"Against all diplomatic ethics, the US ambassador tells me: 'If you don't jail him, you must resign'," Alko told reporters, citing a recent conversation with Eikenberry.
"Do diplomatic ethics allow the attorney general of a country to be threatened in such a manner?" Alko said.
He said his office did not have enough evidence against the banker, Rafiullah Azimi, who was allegedly involved in a corruption case linked to a former minister now living in Britain after eluding an arrest warrant in March.
"The US embassy has a regular dialogue and a strong partnership with the attorney general and his office, including robust mentoring and training programmes," the embassy said in response.
"The ambassador’s discussions with his counterparts are private and we’re not going to comment on them. The United States continues to respect that personnel decisions are for the Afghan Government to make."
Alko also accused the US and Britain -- the main countries providing troops for the war against the Taliban -- of not cooperating with Kabul in arresting Afghan officials living abroad but wanted in Afghanistan on corruption charges.
He said that during his meeting with Eikenberry "I answered calmly: 'If I am to resign I'll submit my resignation to our president or our parliament".
Afghanistan is one of the world's most corrupt countries, according to monitoring organisation Transparency International, which rates it just above lawless Somalia.
A US media report said Monday that billions of dollars in international aid money is regularly shipped out of Kabul on scheduled commercial flights, packed into suitcases, some of it even registered with customs.
A senior US lawmaker on Monday angrily blocked billions of dollars in aid to the war-torn country, vowing not to give "one more dime" until Afghan President Hamid Karzai acts against corruption.
Representative Nita Lowey, who sits on the powerful committee in charge of the budget, said she would hold hearings into allegations that top Afghan officials flew suitcases full of cash from US aid to foreign safe havens.
Alko told reporters Tuesday that the attorney general's office was clean.