Many customers have either already paid bills for May and June or have offered to pay what they owe immediately, a source close to the administrators said.
KPNQwest carries up to 40% of European internet traffic and signed a number of contracts with blue-chip customers this year, before filing for bankruptcy at the end of May.
They include Hewlett-Packard, which signed a three-year, multimillion-pound deal for the carrier to host its European network, Opodo, the major new portal for a consortium of European airlines and Dutch airline KLM.
The trustees want to keep the KPNQwest running this month and possibly through July, giving them more time to find a buyer and customers more time to find an alternative carrier.
It is unclear what impact a shutdown by KPNQwest would have as many customers have already switched to other networks.
Italian internet service provider Tiscali is a major KPNQwest user, while United Pan-Europe Communications group is seeking an alternative route for its Chello broadband internet service in the Netherlands.
WorldCom, BT Ignite, and even Global Crossing Holdings, which has its own financial troubles, are targeting KPNQwest customers.
Parts of the KPNQwest network have already gone down, according to Web portal company Lycos Europe, which is connected to the KPNQwest network in several of its locations.
"I am not sure what happened, whether it was KPNQwest or one of their suppliers, but part of the network was switched off last week," said Frank Weller, chief technology officer for Lycos Europe. "Our data traffic is taking different routes now. We had fallbacks in place."
Dutch company Hit Rail, which manages IT and data communications projects for 14 European railway operators, signed a deal with KPNQwest at the end of March, for a 16-country, 29-site virtual private network and implementation.
Two sites were live when KPNQwest toppled. Hit Rail is now talking to alternative suppliers, said Antonio López, Hit Rail's director of network services.
"We have not received any news from KPNQwest since last Friday. My account manager, who was based in France, was dismissed on Monday. We reopened negotiations with suppliers," he said.