Instead, the system should be led by local public health leaders, who have proved they are more successful in tracing the contacts of infected people, Labour said.
The call comes after it was revealed that just 56 per cent of close contacts handled online or by call centres run by Serco and Sitel, a second private firm used, are being reached.
In stark contrast, 90 per cent were contacted by Blackburn with Darwen Council, one of a growing number of exasperated town halls now setting up their own systems.
That success was put down to local knowledge and a Blackburn phone number, instead of the 0300 number used by Boris Johnson’s “world-beating” centralised system, which provokes suspicion.
Now Labour has written to Matt Hancock, the health secretary, to say Serco’s contract should be paused because it is “ineffective and not fit for purpose”.
“It is our view that no additional funding should be given to Serco as part of this contract,” says the letter, from shadow health secretary Jonathan Ashworth and Rachel Reeves, the shadow Cabinet Office minister.
The pair added: “The stakes are too high to tolerate failure in either the operation or the design of this crucial public service.
“We have long argued that test and trace should be led by local areas, backed up by the resources and national support to make this happen. It is not too late to improve the system and put in place new, locally-led contact tracing measures.”
The Serco deal has been shrouded in secrecy, but Labour said an initial £108m award is due to soar to £410m, with a decision in a fortnight’s time.
The letter also raises questions about the vast number of subcontractors employed by Serco, totalling 85 the party said.
It demands answers on “numerous reports in the press about contact tracers being paid to be effectively idle for large amounts of time, given little work or instruction”.
And it criticises when Serco “revealed the personal email addresses of 296 contact tracers in May”, asking: “How is this acceptable?”
Ministers should publish the performance measures set for the firm and “the breakdown of performance against these indicators”, the letter says.
Mr Hancock has been accused of an obsession with hopes for a contract-tracing app – now abandoned it seems – and for using private firms over experienced public officials.
However, there are signs that ministers are going cool on Serco, its own boss suggesting some of its 10,500 staff will be shed because of a lack of work for them to do.