Greece has lost a lot of trust and euro zone governments must decide whether to expand their risk exposure to Athens, European Central Bank policymaker Jens Weidmann said on Thursday.

One option the Greek government has looked at to fund itself is for the ECB to raise a cap on Athens' issuance of Treasury bills, or short-term debt.

But Weidmann, who is chief of Germany's Bundesbank, said the euro zone's central banks should ensure Greece's banks do not worsen their liquidity position by buying up Greek government debt for which he said there is still no market.

He did not see Greece regaining access to capital markets by mid-year.

"A lot of trust has been lost," Weidmann told a news conference to present the Bundesbank's 2014 results.

Turning to monetary policy, Weidmann said the ECB should have looked through downward cost pressures resulting in significant part from the sharp fall in oil prices.

"Negative inflation rates are only of temporary nature ... I see no signs of second-round effects," he said.