It is thought to be the first time children's services have been put out to tender to such an extent in England, according to the Royal College of Paediatrics and Child Health (RCPCH).
Virgin Care, part of Sir Richard Branson's group, is understood to be one of two profit-making companies bidding for the £130 million, three year contract.
The other is Serco, which already runs out-of-hours services and helps provides community health services for children in neighbouring Cornwall. Also bidding are Barnardo's and other charities.
Among the children's services that form the Devon contract are palliative care for the dying, therapy and respite care for disabled children, treatment for those who are mentally ill, and child protection.
A source told The Guardian that one of the for-profit firms was the favourite to win the tender, which will be awarded to "the most economically advantageous" bid.
However, verifying that is difficult as NHS trusts are reluctant to publicise widely such arrangements because they are so controversial.
Advocates of allowing private firms and charitable organisations to bid to provide core NHS services argue the process of competition will drive standards up and costs down. They also maintain it stimulates more innovative ways of working.
But critics say it makes it much harder for doctors to work collaboratively, and risks putting profit before patient safety.
Professor Terence Stephenson, president of the RCPCH, said: "Children's services are complex, as some tragic high-profile incidents have shown.
"It is essential that any provider has proven expertise in managing not only clinical services but also key areas such as safeguarding and, on the face of it, it does not appear that all of the shortlisted bidders have experience in managing such services."
A spokesman for Andrew Lansley, the Health Secretary, emphasised that allowing private firms to provide NHS services did not amount to privatisation of the NHS. Mr Lansley and the Prime Minister have regularly said the NHS will remain "free at the point of delivery" - free to patients.
The spokesman said: "We support patient choices and whoever is best getting the contracts. We reject the idea that because a private company might get it, it is privatisation."
Virgin Care said it could not comment while the tendering process was ongoing.
Serco has been criticised for its out-of-hours provision in Cornwall, for unanswered calls and low nurse numbers. But Paul Forden, managing director of clinical services, said response times were now "among the best in the country".
Jayne Carroll, from NHS Devon, said it was contracting out services "to secure the best possible service" for children.
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