The offers came from a consortium led by Italy's post office, France's Amundi (>> Amundi S.A.), Australia's Macquarie (>> Macquarie Group Ltd) and U.S.-based Ameriprise Financial (>> Ameriprise Financial, Inc.), the sources said.

One of the sources said that UniCredit would shortlist two bidders by Monday, aiming to take a final decision on the buyer by Dec. 10.

Chief Executive Jean Pierre Mustier, who took the helm at Italy's largest bank by assets in July, is looking to sell Pioneer and other businesses to boost the lender's stretched capital base.

It is also expected to launch a rights issue, which bankers expect to be in a range of 8-10 billion euros. The size of the cash call will depend on how much money UniCredit nets from disposals.

One source said the offers valued Pioneer at between 3.2 billion and 3.4 billion euros, excluding a 325 million euro cash position held by the asset manager.

The second source indicated a value range of 3.7-4.0 billion euros, but it was not clear if this included Pioneer's liquidity.

Pioneer is Europe's sixth-largest asset gatherer and has assets under management of around 225 billion euros.

As investors around the world search for higher returns, asset gatherers have grown rapidly in recent years, making them an appealing target for banks and financial institutions.

The financial wealth of Italian families, excluding property, totals 3.9 trillion euros.

According to 2014 data, only 32 percent of these funds was invested in asset management and insurance products, well below an average in leading European countries - making Italy a country with high growth potential for asset managers.

Poste Italiane (>> Poste Italiane SpA) is the only bidder who officially confirmed on Thursday having presented an offer for Pioneer. It is bidding with Anima Holding (>> Anima Holding SpA) and Italian state lender Cassa Depositi e Prestiti, it said in a statement.

UniCredit confirmed on Friday it was in talks with potential buyers for Pioneer, saying there was no certainty the talks will lead to a deal.

($1 = 0.9184 euros)

(Additional reporting by Francesca Landini in Milan and Maya Nikolaeva in Paris; Editing by Tom Brown/Ruth Pitchford)

By Maria Pia Quaglia and Gianluca Semeraro