Deutsche Post boss Klaus Zumwinkel, who is under investigation for tax evasion, has offered to quit.
Mr Zumwinkel is suspected of tax evasion
Deutsche Post's executive committee has called on the board to accept his resignation, which German Chancellor Angela Merkel described as unavoidable.
The German government is a major shareholder in Deutsche Post, which also owns the global delivery firm DHL.
Germany is cracking down on wealthy businessmen using offshore bank accounts to escape taxes.
Hundreds more Germans face being investigated for tax evasion.
"In a large number of cases, key personalities from our country sought to avoid their duty of solidarity via taxation by practising fiscal evasion to Liechtenstein," a finance ministry spokesman said.
Deutsche Post said on its web site that it was in the interest of the company for Mr Zumwinkel to resign.
Peer Steinbrueck, the German Finance Minister, said the scandal had damaged public morale.
Deutsche Post has a market share of nearly 90%
"If the public has something like this as a role model then they'll start having doubts about this economic and social system. I didn't think this was possible."
On Thursday, the Deutsche Post boss was questioned by police and his home and office were raided.
Prosecutors suspect that he avoided paying 1m euros (£750,000; $1.4m) in taxes using banks in Lichtenstein.
He has not yet made a public statement about the allegations.
John Allan, the British finance director at Deutsche Post, has been tipped by some analysts as a likely successor.
Mr Zumwinkel, 63, is one of Germany's most influential business figures, credited with transforming the postal service into a logistics giant, as well as developing its retail banking arm, Postbank.
He has been chief executive of Deutsche Post for 18 years.
He is one of the longest-serving chief executives of a German blue chip company and also heads the supervisory board of Deutsche Telekom.
But he is no stranger to controversy.
He sold about 5m euros worth of shares after the introduction of the minimum wage for postal workers - which he fought for - boosted the company's share price.