* BPER offers to buy controlling stake in Carige for 1 euro

* Bid contingent on industry-financed fund putting in 1 bln euros

* Fund meets on Thursday, will discuss BPER's offer

MILAN, Dec 15 (Reuters) - Shares in Italy's Carige soared 14% on Wednesday on a takeover proposal by rival BPER which is contingent on the industry first pumping 1 billion euros into the loss-making bank.

Italian banks are stakeholders in the FITD depositor protection fund which owns 80% of Carige following a 2019 rescue that cost them 600 million euros.

The fund has been looking for a buyer for Genoa-based Carige since Cassa Centrale Banca, which holds 8.3% of Carige, walked away from a proposed acquisition in March.

BPER, Italy's fifth-largest bank by assets, said it would take on Carige - paying a nominal 1 euro for FITD and Cassa Centrale Banca's combined 88.3% stake - provided the deal helped it shed bad debts, significantly boosted earnings per share and did not affect its capital.

The conditions echo those set by UniCredit for a rescue of state-owned Monte dei Paschi, although that deal fell apart in October.

To meet BPER's terms, the FITD fund would need to add 1 billion euros to Carige's capital, which would also cover integration costs and credit risks.

The fund's board will meet on Thursday to discuss whether to enter exclusive talks by Dec. 20 as demanded by BPER which wants to clinch an accord by Dec. 31.

A sale would solve one of the industry's problems, but sources have said that FITD members are divided, with some afraid that the cost of rescuing Carige will only rise over time, and others unhappy to fund BPER's expansion plans.

The fund's members have voting rights that reflect their contributions, so heavyweights like Intesa Sanpaolo and UniCredit wield more power.

An acquisition of Carige, which said in July said it needed a further 400 million euros in capital to complete its turnaround and return to profit in 2023, would push BPER's assets above 150 billion euros. That follows a 40% expansion in its assets last year when it bought branches as part of rival Intesa Sanpaolo's takeover of UBI.

"Consolidation in the Italian banking sector is not over," Intesa Sanpaolo analysts said.

BPER has been set on an expansion path by its top shareholder, UnipolSAI.

UnipolSAI boss Carlo Cimbri on Monday said BPER would only pursue merger deals with a strong financial rationale, adding a tie-up with Popolare di Sondrio would make sense.

If the FITD accepts BPER's offer, the bank would launch a mandatory takeover on remaining Carige shares at 0.80 euros each, spending around 70 million euros.

Shares in Carige closed up 13.7% on Wednesday at 0.762 euros. BPER rose 6%.

($1 = 0.8875 euros)

(Editing by David Goodman, Valentina Za, Kirsten Donovan)