* Central bank trimmed Core Tier 1 estimate - report
* Board rejected cap hike in March
* Shares down 0.5 percent, sector slightly higher
MILAN, April 19 (Reuters) - Banca Popolare di Milano's board will discuss a possible capital increase on Tuesday, a source close to the issue said, which newspapers reported could be about 1 billion euros ($1.42 billion).
The mid-tier cooperative lender is the latest Italian bank eyeing a capital increase to meet tougher Basel III capital rules. The Bank of Italy has urged banks to raise capital levels that are low compared with many European lenders.
"The board is meeting today and will be examining the hypothesis of a capital increase," said the source, who spoke on condition of anonymity.
The board will discuss results of a Bank of Italy inspection of its accounts. La Repubblica newspaper said the central bank had shaved its estimate of Core Tier 1 capital by 6 percent after recalculating risky assets.
Corriere della Sera newspaper said the capital increase could be about 1 billion euros or more. The board rejected a proposed 600 million euro capital increase on March 29.
Pop Milano's Core Tier 1 ratio was 7.1 percent at the end of 2010. Core Tier 1 capital mainly groups common equity and retained earnings and is best suited to absorb possible losses.
Four Italian banks -- Banca Monte dei Paschi di Siena, Intesa Sanpaolo, Banco Popolare and UBI Banca -- have announced or carried out this year capital increases totalling 10.5 billion euros.
UniCredit, Italy's biggest bank by market value, has no need for now to ask shareholders for a capital increase, Chief Executive Federico Ghizzoni said in an interview with Il Sole 24 Ore newspaper on Tuesday.
However, the bank will re-evaluate the situation at the end of the year, Ghizzoni said. UniCredit carried out two capital increases totalling 7 billion euros during the global credit crisis.
Shares in Pop Milano were down 0.5 percent at 2.51 euros at 0909 GMT. The STOXX Europe 600 Banks index was up 0.2 percent.
(Reporting by Stephen Jewkes and Ian Simpson; Editing by Erica Billingham)
($1=.7039 Euro)