Achleitner is due to retire next May after 10 difficult years in office that saw two changes of CEO, billions of euros of losses and huge fines.
The 64-year-old has survived three shareholder votes on whether to oust him but said he will go next year.
With less than a year to go, the bank's supervisor, the ECB, is still waiting for Deutsche to identify a successor and has ratcheted up pressure on Germany's biggest bank and its chairman in recent months, the people said.
The ECB's chief supervisor Andrea Enria has addressed this question multiple times in ongoing high-level communication with the bank, said one of the people, who is an official with direct knowledge of the matter.
Deutsche Bank declined to comment.
Deutsche has recently returned to profit but remains under the close watch of regulators who guard the health of the global financial system, of which the bank, as one of the world's biggest lenders, is a critical part.
In particular, they have been concerned by lapses in Deutsche's money laundering controls - a problem that extends back throughout Achleitner's tenure.
In a major setback for Deutsche, Germany's financial watchdog BaFin in April ordered it to enact further money laundering safeguards and extended a mandate for a special monitor to keep track of progress.
The ECB is making the repeated demands because it wants Achleitner to leave in an orderly fashion, with the central bank having enough time to vet the candidate to replace him, the people said.
Three people familiar with the bank's search for a successor said there was a shortlist of three German men, with one adding that Deutsche would be ready to announce a successor by February or March next year. Such a late date would leave little room for manoeuvre should the candidate fall through.
If the men remain the frontrunners, it may also spark questions about lack of diversity as Deutsche seeks to install more female leaders.
German stock exchange chief Theodor Weimer was the preferred candidate, the three people said.
Weimer, however, has publicly dismissed the idea saying earlier this year that his taking the post was "not at all a topic of discussion". A spokesperson for Weimer declined to comment further.
It is unclear whether Weimer could change his mind. His contract as CEO of Deutsche Boerse was extended last year to 2024. Regulators would not approve his keeping both posts, the official with direct knowledge of the matter said.
Furthermore, accepting the position at Deutsche Bank would require him to resign from a job that paid him 6 million euros last year - a multiple of Achleitner's earnings.
Also in the running, the three people said, are Norbert Winkeljohann, an accountant and chair of Bayer, and Frank Witter, a new board member who was until recently Volkswagen's finance chief.
Winkeljohann did not respond to a request for comment and Witter declined to comment.
Achleitner has privately sought to reassure investors that the search is underway. In recent weeks, he told investors that the new chair's name will very likely begin with the letter 'W', according to one person with knowledge of the matter.
But the vague nature of his plans has left regulators, and some investors, frustrated.
"The quicker we have clarity over a successor the better," said Vanda Rothacker of Union Investment, a shareholder in the group.
Regulators have struggled for years to rein in Deutsche Bank following numerous scandals and governance issues.
The U.S. Department of Justice has investigated the lender's role https://www.reuters.com/article/us-deutsche-bank-money-laundering-exclus-idUSKBN1Y60ED in the 200 billion euro ($220 billion) Danske Bank money laundering scandal, people familiar with the inquiry have told Reuters.
Deutsche has also had to pay nearly $700 million in fines by U.S. and British regulators in a separate money laundering case involving $10 billion in so-called mirror trades from Russia.
The bank has said it has significantly improved controls in recent years.
Earlier this year, the ECB clashed with CEO Christian Sewing, demanding he relinquish day-to-day oversight of Deutsche's sprawling investment bank, fearing he had too much on his plate.
Sewing, who eventually complied, for his part has sharply criticised the ECB's ultra-loose monetary policy that has required lenders to pay to deposit money with the central bank and eroded profits.
(Editing by John O'Donnell, Kirsten Donovan)
By Tom Sims and Patricia Uhlig