Around Â£9 billion of newly-created equities from Barclays, Bradford & Bingley and the UK's biggest lender HBOS remain unsold - and might now have to be purchased by the cash calls' underwriters.
Recent falls of bank stocks to below the rights issue prices have impacted on sales - with customers able to buy the shares more cheaply on the open market.
For example, HBOS closed in London at 254.5p yesterday, while offering rights issue shares at 275p.
However, a last-ditch meeting of City fund managers today before HBOS' deadline tomorrow might lead to at least some of the lender's equities being sold, the Times reports.
Heavy exposure among the funds to tracker products, which lose and gain on the performance of indices as a whole, is thought to be likely to swing the balance for some towards selling - with further gloom in the financial sector likely to lead to drops in markets, and erosion of the trackers' value.
"It's a decision you have to makeâ¦if you are exposed to banks through a FTSE 100 index tracker, you have to decide whether to allow that position to be diluted," an unnamed manager told the newspaper.
"If you are an active manager, you can take more of a view on whether you're happy about reducing that exposure."