Both Bank of Ireland and Davy have declined to comment on a report that the bank has made an exploratory approach to the embattled stockbroker about the possibility of buying it.
The Irish Times reported that high level contact has been made between the two firms, as Davy tries to contain the fallout from its record €4.1m fine from the Central Bank of Ireland last week.
The sanction was imposed on Ireland's largest stockbroking firm for regulatory breaches uncovered around the 2014 sale of a bond by a client, who was not told that the buyer was a consortium made up of 16 Davy staff.
On Saturday, three senior figures at Davy, including chief executive Brian McKiernan, resigned from their positions.
Then on Monday the National Treasury Management Agency withdrew Davy's authority to act as a primary dealer in Irish Government bonds, ahead of a planned auction of up to €1.5 billion in bonds tomorrow.
Davy subsequently closed its bond desk, making four staff redundant in the process, and said the decision meant none of the 16 people who were part of the 2014 consortium work for it any longer.
However, Davy is owned by management and employees, after they bought it from Bank of Ireland in 2006 for around €350m.
Some of those members of Davy's senior management who have departed the firm in the last week remain large shareholders in the business, something Government and opposition politicians have expressed concern about.
A takeover of the firm would, therefore, provide a mechanism for those individuals to cut ties completely with the business.
The Irish state still owns 14% of Bank of Ireland.
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Davy currently acts as corporate broker to Bank of Ireland and last week, on foot of the Central Bank fine, the bank said it was disappointed that Davy had breached its regulatory obligations.
Last week, AIB which is 71% owned by the state, confirmed it was buying Goodbody Stockbrokers for €138m.
Under the deal, Goodbody will be allowed to maintain its current remuneration policy, which allows for bonus and salary levels not allowed in AIB under Government rules.
Govt to publish legislation to give regulators more powers - Taoiseach
Sinn Féin Leader Mary Lou McDonald claimed the toxic culture that brought this country to its knees more than a decade ago is still alive.
"This recent scandal, (Davy) is not an isolated issue," she said.
She told the Dáil that it was public anger which brought about resignations at Davy.
But she said people need to know where all the 16 investigated in the Davy now work.
The Taoiseach replied saying the actions of these Davy executives was unacceptable as was their efforts to frustrate the Central Bank investigation.
He said the Central Bank has extensive powers to sanction individuals but Government will soon publish legislation to give regulators more powers.
The Taoiseach added that everyone in Government condemns what happened and was determined to get the proper authorities to deal with it.
It is up to the Director of Corporate Enforcement regarding further action, he said.
Micheál Martin also said Sinn Féin needed to be more accountable about its party finances describing some of its funding as a "a shady enough transaction that would make even a stockbroker blush".
He accused the party of "cozying" up to high finance when it suits them.