Father Fidelis Mukonori said he would provide “advice” as an elder statesman, including to the new president.
Mr Mugabe, 93, resigned on Tuesday after a military intervention and days of mass protests.
Mr Mukonori said he could not confirm reports that the ex-leader was granted $10m (£7.5m) to ease him out of office.
Emmerson Mnangagwa was sworn in to replace Mr Mugabe as president on Friday.
Mr Mnangagwa, long a close ally of Mr Mugabe, was sacked earlier this month, triggering the political crisis that eventually saw his boss’s downfall.
Father Mukonori, 70, who is close to Robert Mugabe and acted as a mediator between him and the military, said the new president would go to his predecessor for political counsel.
“In the African world, senior citizens are there for advice,” he told the BBC’s Richard Galpin at a church outside the capital, Harare, after leading a service that included prayers giving thanks for the peaceful transfer of power.
He referred to what Mr Mnangagwa said about his predecessor at his inauguration.
“When he says ‘he’s my father, he’s my leader, he’s my mentor’, you tell me he’s going to stay off from his father, from his mentor, from his leader? I don’t think so.”
The priest said that Mr Mugabe and his wife Grace remained at their house in Harare and had no plans to leave the country.
The military takeover came in response to Mr Mugabe’s decision to position Grace as his successor and sack Mr Mnangagwa from the vice-presidency.
Father Mukonori said he could not confirm reports that the ex-president was granted millions of dollars and promised that his assets would not be touched to persuade him to step down.
“We didn’t offer him anything… He resigned for the good of Zimbabwe,” he said.
He added: “What I have read in the newspapers is about immunity [from prosecution], and that he will be looked after like any other former head of state.”
Mr Mugabe leaving power, he added, was the best thing he had ever done.
Separately on Sunday, Robert Mugabe was described as being “quite jovial” by a nephew in an interview with the French news agency AFP.
“He is actually looking forward to his new life – farming and staying at the rural home. He has taken it well,” Leo Mugabe said.
He said that Grace wanted to focus on already announced plans to build the controversial $1bn Robert Mugabe University in Mazowe, near Harare.
There are fears that President Mnangagwa, who is associated with some of worst atrocities committed under the ruling Zanu-PF party since independence in 1980, will not usher in the democratic reforms that many in Zimbabwe are hoping for.
But Father Mukonori said he believed the former spymaster knows that democracy is “crucial”.