The White House is looking for monetary policy expertise for the position, according to the sources and Lindsey would fit that bill. He was a governor of the central bank from 1991 to 1997.
Lindsey, president and CEO of economic consultancy firm The Lindsey Group, is also interested in exploring the Fed vice chair job, according to the sources.
The economist was director of the National Economic Council under Bush. He also served in the administration of former President George H.W. Bush and under President Ronald Reagan. Lindsey was also heavily involved behind the scenes in crafting the recent tax cut legislation.
The vice chair seat was vacated in October by Stanley Fischer for personal reasons. President Donald Trump's nominee for Fed chairman, Jerome Powell, still needs to be confirmed by the full Senate, but the Senate Banking Committee approved his nomination earlier this month. The term for the current chair, Janet Yellen, expires in February.
Last month, Mohamed El-Erian, chief economic advisor at Allianz, was reportedly being considered for the position, CNBC and other news outlets reported.