Teams including Bovis, Jarvis and developer Godfrey Bradman were outraged last month when they were told they would have to rebid for the £60m contract, three years after it was launched.
Now the Home Office has agreed to pay a substantial proportion of the extra cost.
The new bids will assume one move for staff rather than the two specified in the first bid. Bidders expect the Home Office to pay them up to £1m each to rebid on the basis of this "single decant", which will involve designing a new building.
The Bradman team has already designed a building for the department, located at the former site of the DOE at Marsham Street, Victoria. But it is understood to believe its bid will need further work, and it too wants extra funding.
Other bidders are also expected to design a new building for the Home Office for Marsham Street. With the fees issue settled, the competition is expected to go ahead. Some bidders still want their costs from the earlier round of bidding, estimated at £2m each, refunded.
The two bidders now required to design a new building are also understood to be aggrieved about the timescale.
Although Bradman, working with Mace and architect Terry Farrell & Partners, took over two years to design his new-build project, Jarvis and Bovis will have just six months.