* M3 annual growth rate in Nov 3.8 pct vs 3.9 pct in Oct
* Consumer credit, loans to businesses fall in Nov
* Weak loan data puts pressure on ECB to act (Recasts with loan detail, analyst comment)
FRANKFURT, Jan 3 (Reuters) - Loans to companies in the euro zone extended their fall in November, underlining the bloc's economic weakness and increasing pressure on the European Central Bank to cut interest rates further.
On the year, loans to the private sector fell by 0.8 percent in November - the same drop seen a month earlier. A breakdown of figures released by the ECB on Thursday showed a drop in consumer credit and loans to businesses.
"In part, the weakness in lending undoubtedly stems from low demand from the non-financial private sector," said Howard Archer, economist at Global Insight.
"Nevertheless, the concern is that a number of companies who do want to borrow ... and are in decent shape are finding it hard to, so tight credit conditions are handicapping euro zone growth prospects," he added.
The release of the money supply data reinforced a bleak economic picture in the 17-country euro zone, where a slowdown in factory activity deepened in December. German unemployment also rose for the ninth month running in December.
The ECB's own forecasts suggest the euro zone economy will shrink 0.3 percent this year, and the bank's policymakers discussed cutting interest rates last month before opting to hold them at a record low of 0.75 percent.
The money supply data showed that consumer credit dropped by 3.1 percent on the year in November, down from a 2.9 percent fall a month earlier. Loans to non-financial corporations fell 1.4 percent, after a drop of 1.5 percent in October.
On a more positive note, annual growth in deposits placed by households and non-financial corporations accelerated - to 3.8 percent and 4.2 percent respectively - offering the ECB some comfort.
"Shrinking lending to companies may be reversed as the confidence rebound inspired by the ECB's safety net as well as improved euro zone crisis management spreads from financial markets to the real economy and as austerity starts to ease at least in some countries," said Berenberg Bank economist Christian Schulz.
The ECB has taken some of the heat out of the euro zone crisis with its new bond-purchase programme - dubbed Outright Monetary Transactions (OMT).
But some analysts say the bleak economic outlook means there are grounds for another rate cut, which Archer expected in the first quarter.
ECB President Mario Draghi said after the ECB's December policy meeting there had been a "wide discussion" about rates, initially fuelling expectations the bank could cut again.
But Joerg Asmussen, an ECB Executive Board member, said late last month he would be "very reluctant" about the ECB cutting its deposit rate - now at zero - any further, adding that "our (monetary) policy is very accommodative".
Another board member, Yves Mersch, said he did not see the logic of a debate about the ECB cutting its main rate from a record low of 0.75 percent. A third board member, Peter Praet, said earlier last month there was little room to cut.
"The commitment to potentially unlimited bond purchases (OMT) is the key policy tool of the ECB," said Schulz.
"To ensure its credibility, ECB President Draghi will have to ensure maximum support for it in the Governing Council, which gives hawks a disproportionate weight and will probably prevent another rate cut to support the economy," he said. (Reporting by Paul Carrel, Eva Kuehnen and Sakari Suoninen; editing by Stephen Nisbet)