One possibility is that the investment bank ceases to exist as a separate division at some stage.
State Street's pay and employee benefits declined by 15% during the second quarter.
The Swiss financial giant has changed its entire executive team and half its board of directors in the past 18 months.
The Zurich-based bank told investors last month to expect a third straight quarterly loss.
Khan was given the regional leadership role for Europe, the Middle East and Africa at UBS in 2021.
“It's unrealistic and it is not what employees want,” he said.
The Zurich-based bank reported net income of $2.1 billion, compared with analyst estimates of $1.9 billion.
Credit Suisse last week shocked investors with its fifth profit warning in six quarters.
Wealthy individuals invested in Russian assets are facing frozen funds and demands for more collateral.
The bank is emerging from one of its most turbulent periods since the financial crisis.
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