According to Channel 4 News, both the Department for Transport (DfT) and the Treasury were asked to assess the potential benefits and rewards of the project during the Tory leadership contest earlier this year.
In response, the DfT has already produced a "factual paper on the subject", says Channel 4.
It adds that Johnson asked officials about "the risks around the project", which may include "WW2 munitions in the Irish Sea".
The idea of a bridge connecting Northern Ireland and Scotland was first floated by Johnson last year.
A statement from a government spokesman at Number 10 said: "Government regularly commissions work to examine the feasibility of projects.
"During the leadership campaign, candidates spoke about a number of issues which resulted in Number 10 commissions ahead of a new Prime Minister taking over.
"This PM has made no secret of his support for infrastructure projects that increase connectivity for people and particularly those that strengthen the Union."
The Times reports that Alan Dunlop, visiting professor of architecture at Robert Gordon University in Aberdeen and the University of Liverpool, "has drawn up two sets of plans on which any proposal is likely to be based".
The newspaper says that one is a crossing from Larne in County Antrim to Portpatrick in Dumfries and Galloway, which would cost about £20 billion.
The second proposed bridge would link the Mull of Kintyre and Torr Head in Antrim at a cost of up to £15 billion, it adds.
The bridge is the latest pf several large-scale infrastructure projects that Johnson has expressed support for, including the now-scrapped central London garden bridge across the Thames, an airport in the Thames Estuary, and a bridge from England to France.