The move brings Italy's top insurer a step closer to controlling its smaller rival after Generali announced a 1.17 billion euro ($1.36 billion) buyout offer on Cattolica in May to further strengthen its domestic market leadership.
The Trieste-based company became Cattolica's single largest shareholder with a 23.7% stake last year, relegating Berkshire Hathaway to second place when excluding the 12.3% held by Cattolica itself.
Buffett's Berkshire Hathaway has tendered its 6.93%, the source said, confirming an earlier report by news agency Radiocor.
So far, 24.77% of the shares subject to the offer, or 18.9% of the total capital, have been tendered in the bid, which ends on Oct. 29, Borsa Italiana said in a statement without giving further details.
That implies that Generali would hold at least 42.6% of Cattolica, nearing the 50% plus one share threshold that the company has set as a condition of the offer.
($1 = 0.8601 euros)
(Reporting by Andrea Mandala and Gianluca Semeraro, editing by Louise Heavens, Kirsten Donovan)