All depends on the method of procurement.
If it's Design & Build then the chances are the tendered costs will be significantly higher than £300 million, but with City having a degree of certainty with the costs, barring 'unforeseen' extras that could not reasonably have been allowed for.
If it's a Traditional JCT Contract then the tendered price will be much lower, but with the very real and, in fact, probable, chance of costs escalating and then spiralling out of control.
Large construction projects seem to be moving towards an NEC Contract, these days, whereupon the two parties agree a target cost, say £300 million, and then if the works actually cost £350 million, both City and the Contractor share equally the £50 million 'hit'; i.e. City pay £25 million and so does the Contractor. However, if the costs actually came in at £250 million, they both share the saving, so City would pay them £25 million above the £250 million as a nice little bonus for keeping the costs to £50 million below the target cost. It works really well, because it incentivises the Contractor to keep costs down rather than escalate them.
