In a letter, the prime minister and cabinet secretary Simon Case also warned ministers and permanent secretaries not to leak private conversations.
It comes after Johnson overruled independent advice that Patel had breached the ministerial code by “shouting and swearing” at Home Office staff, sparking the resignation of the PM’s adviser on ministerial interests Sir Alex Allan.
In a letter to all ministers and permanent secretaries, Johnson and Case insisted “never has it been more important that we work as one team”.
They wrote: “Given the unprecedented challenges we currently face as a nation, relationships of mutual trust and respect between politicians and their officials are paramount.
“This includes keeping internal conversations private.
“We should all be able to speak freely and honestly about matters of state.
“We should also feel able to speak constructively about things that are not working, so that we can fix them together promptly.
“There is a particular duty on ministers and permanent secretaries to create jointly across government a culture which is professional, respectful, focused and ambitious for change and in which there is no place for bullying.”
It comes after HuffPost UK revealed that less than a quarter of Tory MPs on Twitter backed Patel with supportive posts despite Johnson’s call for colleagues to “form a square around the prittster” last week.