Germany, the only large euro zone country with a top-notch credit rating, is where the problem is at its most severe.
With the ECB now owning more than a quarter of all outstanding German bonds, funds pay up to 1.5 percent to borrow a 10-year Bund, up from some 0.40 percent a year ago, according to Icap data.
This is putting a strain on investors as they face increasingly frequent demands to put up cash or liquid collateral against their derivative positions due to new regulation.
"If a pension fund can't borrow a bond in time, it may have to sell its own cash bond, foregoing a potential return in the future to fulfill a short-term obligation," Godfried DeVidts of the International Capital Market Association industry body said.
"So basically the pension funds are getting poorer and the pensioners too."
But any decision would then have to be implemented by national central banks, which own the bulk of the debt bought by the ECB and bear the risk for their own bond-lending schemes.
This means the most radical proposals may run into resistance, the sources said.
Both the Bundesbank and the ECB have already taken some steps towards making their bonds easier to borrow.
In late September the Bundesbank started to lend out German government debt directly to dealers, rather than only via its agent.
But for the moment such loans are only extended in exchange for other German debt, limited to a week and subject to a number of constraints.
Last week the ECB said it would give borrowers more flexibility in deciding when they settle their loans, in a bid to limit the number of fails.
"(The change) can help prevent settlement fails in the market, as counterparties can borrow to cover short positions that they only know about on the value date," an ECB spokesman said.