He is believed to have received assurances that the transfer embargo currently imposed on Birmingham will be lifted shortly.
The club were hit with the embargo after filing their accounts late, while their chairman Carson Yeung is also facing five charges of money laundering in Hong Kong.
Players could also be sold this summer to raise funds – midfielder Jordon Mutch has already left for Cardiff in a £3m deal while keepers Jack Butland and Ben Foster, defender Curtis Davies and midfielder Nathan Redmond are also likely to go.
Clark becomes the fourth new managerial appointment in the West Midlands this summer after Lambert, Steve Clarke at West Brom and Stale Solbakken at Championship side Wolves.
He will be the third man to boss both Town and Birmingham after Lou Macari and Steve Bruce.
Macari was at the helm in 1991 and bossed Town between October 2000 and June 2002 after succeeding Bruce, who had been manager from June 1999.
Bruce bossed Birmingham from December 2001 to November 2007, twice leading them to promotion to the Premier League.
Birmingham’s other former bosses include ex-Town winger Pat Beasley, who led Blues to the European Fairs Cup final in 1960, when they lost 4-1 to Barcelona over two legs.
Originally from Stourbridge, Beasley played for Arsenal before a £750 transfer to Town in October 1936.
He made 123 appearances and scored 27 goals in three seasons at Leeds Road, making his sole England appearance against Scotland at Hampden in 1939.