Green Party MEP Dr Molly Scott Cato and lawyers at the Good Law Project wrote to the Brexit department on Thursday, demanding the release of 50 studies on the impact of Brexit on UK industries.
They threatened that if Mr Davis failed to release the studies within two weeks they would begin judicial review proceedings before the High Court.
Mr Davis commissioned the 50 Brexit impact studies earlier this year as the Government prepares to leave the EU.
Downing Street believes that publishing the documents could harm its negotiating position in talks with the EU27.
Dr Scott Cato has already unsuccessfully attempted to force the Government to release the documents through freedom of information requests.
More than 120 MPs have also signed an open letter demanding that Mr Davis publish the findings, accusing the Government of keeping “not only parliament but the public in the dark”.
Jolyon Maugham QC, who runs the Good Law Project, said he believes there is a legal duty to publish studies into how Brexit will affect jobs and living standards.
He said: “It seems to me the Government’s reluctance to release these studies is born not of its ability to damage our negotiating position but what’s politically expedient.”
Mr Maugham was previously involved in the successful Article 50 case brought by Gina Miller, which challenged Theresa May’s ability to unilaterally trigger Brexit, forcing a parliamentary vote on starting the withdrawal process.
It is not yet known which industries are covered by the impact assessments. The Brexit department has vowed to release a list of sectors discussed in the studies “shortly”.