The city council has agreed to consolidate its offices within the modernised building complex, making the scheme the biggest office prelet in the city centre for 20 years.
Sources now tip Shepherd as favourite to sign the contract with view to starting work on the 1960s office refit in the spring.
The job also involves building a 6-storey extension for the council to move into in February 2017.
The new Merrion House will offer 170,000 sq ft of office space, including 50,000 sq ft in the new extension.
Leader of Leeds City Council Councillor Keith Wakefield said the deal to consolidate council offices in one building would bring savings estimated at £15m a year and drive forward the regeneration of the newly-created Arena Quarter.