The expected appointment follows the death of the previous representative, Richard Holbrooke, last December.
Secretary of State Hillary Clinton is expected to formally announce the appointment later this week.
Mr Grossman has spent almost three decades working for the US state department.
His career includes spells as US ambassador to Turkey and assistant secretary of state for Europe.
He helped direct US participation in the Kosovo war and after 9/11, marshalled international diplomatic support for the "war on terror".
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NATO 3.0
The Atlantic Alliance Transforms to Meet 21st Century Threats
