Government Procurement Services, which represents the NHS bodies and local authorities, as well as educational establishments, put the tender out on 31 January, with a deadline of 12 March.
The contract is worth between £100m-£150m over four years, with one lot valued at £80m-£120m and a second lot £20m-£30m.
However, the government has specified that it wants a maximum of ten participants to take the contract on, ruling out a large number of printers.
BAPC chairman Sidney Bobb told PrintWeek that none of his members would be interested in the tender.
He said: "You don't want one contract taking up more than 10% of your turnover, so how big do you think you have to be to take this on? Do the maths. Even if you take the contract at £100m – the minimum – if ten companies take that on, they take on £10m each. That is £2.5m per year.
"Look at the PrintWeek Top 500. How many companies does that leave out in the cold? How many could sustain a contract of that size?"
Bobb added that he did have some sympathy with government, which he said would face more in the way of costs if contracts were broken up.
But he added: "It is annoying because when these contracts come out, even the incumbents don't stand a chance.
"It will go to a print management company and if one of my members is in bed with a print management company, they might see some of the work. If not, they have no chance. The print won't be any cheaper; the management will."