The Most Rev Justin Welby has “reached out a number of times” without success according to The House, Parliament’s magazine.
A spokesman for the archbishop confirmed the approach to The House.
He said: “The archbishop would be happy to meet the Home Secretary to discuss issues of mutual interest and concern. In the past, the archbishop has met other home secretaries. It is not unusual.”
A former senior advisor to the 26 bishops who sit in the House of Lords told the magazine the Home Secretary’s response had caused “consternation” and had been seen as a “big slap in the face” at Lambeth Palace.
Government sources said an informal approach for a meeting had been made, according to The Daily Telegraph, which said it is understood the archbishop had met immigration minister Robert Jenrick.
The archbishop and other bishops have criticised the government’s plan to send migrants to Rwanda and he condemned “harmful rhetoric” about refugees in a House of Lords speech.
In her latest comments, the Home Secretary predicted there will be a “disintegration in our society” if action is not taken to curb the number of migrants arriving in Britain via small boats.
Speaking to GB News, she repeated an argument made in her speech on migration in the United States this week that the threshold for claiming asylum needs to be raised.
Her comments indicate she does not plan to back down on her calls for the international reform of refugee rules — such as the United Nations’ Refugee Convention — despite facing backlash, including from her own party.
In a speech to a centre-right think tank in Washington DC, Mrs Braverman warned the West faced an “existential” threat if countries were not able to control their borders from unauthorised arrivals.
People want control of their borders and they want government to get a grip on this and if we don’t it will lead to a disintegration in our society
Senior Tory MPs have criticised Tuesday’s speech, singling out her comments arguing that being discriminated against for being gay or a woman should not grant a person refugee status in the UK.
At least three ministers are among those who have complained, according to The Times, with the newspaper saying Chief Whip Simon Hart is expected to speak with the Home Secretary to convey the upset in the Tory ranks before their party conference starts in Manchester on Sunday.
In an interview with the deputy Tory chairman for his programme Lee Anderson’s Real World, Mrs Braverman said migration was an international crisis that requires a “global solution”.
She said: “Whether it is New York City, whether it is the border with Texas, whether it is Italy, whether it is the Channel, we need to look again at whether these international rules are fit for purpose.
“And what I said in my speech is that there is a lot at stake. There is democratic legitimacy which is at stake.
“People want control of their borders and they want government to get a grip on this and if we don’t it will lead to a disintegration in our society.
“And we need to change — we need to change some of the definitions relating to refugee persecution. It needs to be a high bar if someone is coming to our country fleeing persecution, not a low bar.”
She said it was “reasonable” to question whether accords such as the Refugee Convention, which 149 states have signed up to, are still fit for purpose.
“I don’t think they are,” she added.
The UN’s refugee agency, hitting back at Mrs Braverman’s Stateside speech, has denied there is any need for reform or a relaxation of the definition of a refugee.