According to six sources familiar with the situation, Prime Minister Theresa May's office has asked companies listed on the blue-chip FTSE 100 index to put their name to a public letter welcoming the government's efforts to make Brexit a success.
Miss Fernandes, chairman of group of MPs that circulated the letter, said last night that it was created to show "support for the Government's position".
Reports are circulating that blue-chip directors were unimpressed by attempts to get them to sign the letter, written by government strategists, which praises Theresa May's attempts to secure a transition period around the common market after the United Kingdom leaves the European Union in March 2019.
The letter asserts that the signatories back the government's plan to negotiate an interim period after the formal exit date.
One source who has seen the draft said it was reminiscent of the "Project Fear" campaign that pro-remain supporters were accused of conducting before the European Union referendum.
The letter reads: "We write as leaders of some of the UK's most dynamic businesses operating in sectors as diverse as technology, financial services and advanced manufacturing".
During the European Union referendum campaign, Ms Davidson was a staunch Remain supporter and famously clashed with Vote Leave figurehead Boris Johnson in a TV debate.
The letter said: "We believe this is a good time for employers to work with government and Parliament to make a success of Brexit and secure a bright future for our country".
"This month the Government's Repeal Bill will initiate a programme of legislation that will make Britain ready for life outside the EU".
It says that businesses with existing trade links with countries such as the US, India, Japan and Mexico should "stand ready to use their expertise and networks to cement future relationships" as the United Kingdom seeks new partners outside the EU.
"We have been clear that one of our aims is making sure we deliver a Brexit which is as smooth as possible for business and that gives business the confidence that they need going forward", the spokesman said.
In an astonishing intervention by a senior Conservative casting doubt on whether Brexit will be a success, Ms Davidson criticised United Kingdom government ministers for failing to prepare the public for the uncertainty of hard, drawn out talks with the EU.
However, no such letter appeared from Tory-supporting bosses during this year's general election campaign, reflecting Mrs May's perceived disdain for enlisting the support of company bosses.
But a senior MP said the letter they were accused of backing had been "unacceptable" and was part of a plot to "undermine" attempts by Chancellor Philip Hammond to avoid a cliff-edge Brexit.