Newsnight understands the prime minister personally tried to persuade Mr Dacre to "cut him some slack" in a private meeting in his No 10 Downing Street flat on 2 February, the day European Council President Donald Tusk unveiled details of the deal negotiated by Mr Cameron for the UK.
Mr Dacre told Mr Cameron he would not temper his editorial line on Brexit because he had been a committed Eurosceptic for more than 25 years and believed his readers were too.
In early March, Mr Dacre was told by a Westminster source that the prime minister had tried to persuade Lord Rothermere, a strong supporter of the UK remaining in the EU, to sack him.
The Daily Mail editor was said to be "incandescent" and his resolve to campaign for Brexit "stiffened".
The Daily Mail's sustained and strident campaign for Brexit is regarded by some observers as a significant factor contributing to June's historic vote to leave the EU.