The fund has hired the law firm Morgan, Lewis, & Bockius to conduct the investigation, Murray said.
The firm is expected to report to the IMF executive board on the findings of the investigation by the end of this month, he said.
The news comes as IMF focuses its efforts on trying to help emerging markets in developing countries withstand the worldwide financial crisis.
'No special treatment'
The investigation was first reported on Saturday by The Wall Street Journal, which identified the woman involved as Piroska Nagy, now with the European Bank for Reconstruction and Development in London.
A spokesman at the EBRD in London said he was unaware of the inquiry.
In Washington, an attorney for Nagy, Robert Litt, said that she received no special treatment of any kind, either favourable or unfavourable, and that she left voluntarily in August.
Litt said Nagy does not comment on her personal life.
Murray declined to comment on the details of the allegations against Strauss-Kahn, except to say that the fund is exploring his "relationship with an employee and whether there were any benefits from that relationship".
Strauss-Kahn, 59, said in a statement: "With my full support the IMF is examining an incident which occurred in my private life in January 2008. I have co-operated and am continuing to co-operate with outside counsel to the fund concerning the matter.
"At no time did I abuse my position as the fund's managing director. I look forward to the report of outside counsel."
Strauss-Kahn was written off by critics in 2006 after he failed to win the Socialist Party's nomination for the French presidential election and looked set to finish his career as an economics professor in Paris.
But he was nominated as managing director of the Washington-based global financial institution by his former political rival Nicolas Sarkozy, the French president.
Strauss-Kahn rebuilt a career that appeared doomed in 1999 when he was forced to resign from the Socialist government of Lionel Jospin after he was caught up in a corruption scandal.
A court later cleared Strauss-Kahn.
A fan of classical music and art, as well as a keen skier, Strauss-Kahn has also gained a reputation as a ladies' man.
His marriage to the TV interviewer Anne Sinclair has guaranteed him even more tabloid coverage.