The battle for control of Italy's biggest banks entered a new phase after Banca Monte dei Paschi di Siena launched bids for peers Banco BPM and Banca Generali, valued together at nearly $40 billion.

Monte dei Paschi—widely considered the world's oldest bank still in operation—said Friday that it was making the all-share bids, even as it fights off a takeover offer from larger rival Intesa Sanpaolo.

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The move is the latest twist in a continuing round of dealmaking by Italy's biggest banks. Near death a decade ago, Italy's banking system went through a painful period of bailouts and consolidation that left the market dominated by Intesa and UniCredit.

The stock market value of Italy's banks has soared in recent years as the lenders nursed themselves back to profitability. 

Italy's government has said it wants to see a third national champion in banking, after Intesa and UniCredit, and doesn't want Monte dei Paschi to be broken up. By casting itself as a domestic behemoth under the proposed three-way tie up, Monte dei Paschi improves its position to be more than a pawn.

Monte dei Paschi said the enlarged group would become Italy's third-largest by total assets with a pro forma balance sheet of around 466 billion euros, or $545 billion.

Monte dei Paschi, Banco BPM and Banca Generali have been jostling for the best—or the best-available—partners to complete this next round of empire building. Their aim is to shore up market share across regions and offer more products to savings-rich Italians, while streamlining operations by closing extra branches and combining systems.

Monte dei Paschi said it wants to create a business with capabilities across banking, advisory and wealth management, with more than €810 billion in total financial assets. The move comes as Monte dei Paschi is seeking to fend off a cash-and-stock takeover offer launched by Intesa in June.

Earlier this month, Monte dei Paschi said it was weighing options to preserve its independence, as Chief Executive Officer Luigi Lovaglio noted concerns about the price offered by Intesa and other risks.

The Intesa bid triggered Italy's passivity rule, which means Lovaglio will need support for his plan from two-thirds of shareholders to proceed.

Monte dei Paschi will call a meeting on Oct. 29 to ask shareholders to approve the takeover plans, it said.

For the offers to be successful, Monte dei Paschi will also need to convince investors in Banco BPM and Banca Generali. These include the French bank Crédit Agricole, which owns almost 30% of Banco BPM, and Italian insurer Assicurazioni Generali, the majority owner of Banca Generali.

The bank sees projected annual run-rate synergies of around €2.6 billion. This includes €800 million from the integration of Mediobanca, in line with guidance.

In the deal unveiled Friday, Monte dei Paschi would buy up to 1.52 billion shares of Banco BPM and offer 1.567 of its shares for each Banco BPM share. It is offering 6.958 of its shares for each Banca Generali share.

The offers value Banco BPM at €25.3 billion and Banca Generali at €8.72 billion for a total of €34.02 billion. Monte dei Paschi said the Banco BPM price implies a combination of equals at market, while the Banca Generali offer represents a 10% premium to the bank's closing price on Wednesday.

Monte dei Paschi said it would distribute €4 billion to shareholders as an extraordinary distribution, to be paid with €1 billion in cash and €3 billion in Assicurazioni Generali shares. This represents around 4.5% of Generali's share capital.

Earlier this month, Lovaglio said the bank saw its 13.3% stake in Assicurazioni Generali as "nice-to-have" but was aware that it had attracted interest from a number of market participants.

The offers are expected to generate around 11% accretion in 2028 earnings per share, the bank added.

Monte dei Paschi said it expects the offers to be completed by mid-February.

Write to Michael Hennessey at michael.hennessey@wsj.com and Margot Patrick at margot.patrick@wsj.com

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