It is one of the first UK towers to incorporate design changes as a result of the 11 September World Trade Centre attack. The changes include extra-strong joints and wide stairwells.
Bovis and Mace are also believed to be in the running for a studio in Brentford, west London, for television broadcaster Sky.
Market sources claim that a shortlist of firms is to be drawn up today for the £15m refurbishment and extension job.
Building revealed in September that architect Aukett had been enlisted to draw up plans for the scheme, which will create state-of- the-art studios for Sky News and Sky Sports by next year.
A shortlist of four contractors has also been drawn up for a £16m contract in Brentford, west London, for car manufacturer Audi. They include Carillion, Skanska and Exterior. The client intends to build a regional HQ for the German group.
The 6000 m2 Wilkinson Eyre-designed building includes displays, workshops, administration and conference facilities.
Work on the Audi regional HQ is scheduled to start next spring. The project team includes QS Davis Langdon & Everest, structural engineer Expedition Engineering and services engineer Akeler 10.