Unison has dismissed Lord Mandelson's warning that the trade union movement is leading the left into a blind alley over its attack on Progress, the New Labour thinktank, as it labelled the pressure group an "intolerable" influence on the party.

Dave Prentis, general secretary of Britain's largest public sector union, said the peer's intervention in the row at the weekend was "desperate".

Lord Mandelson has warned that unions are leading Labour back to the divisive era of the 1980s by seeking to exile Progress from Labour, amid plans by the GMB union to lodge a motion against the organisation at the party's annual conference in September.

In his keynote speech at Unison's annual conference in Bournemouth, Prentis rounded on the former business secretary for inferring that the GMB, Unison and Unite – the UK's biggest overall union – are stuck in the past. "And to whoever dug up Lord Mandelson at the weekend, I say, you must be desperate … Mandelson's defence of Progress was ironic. It is a secretive organisation operating inside the Labour party."

Echoing criticisms levelled against Progress by Paul Kenny, the GMB general secretary, he added: "[Progress is] a party within a party, funded by external interests. An influence we will not support. It is intolerable that they should act in that way within our Labour party." Unison is one of Labour's biggest donors, with its members giving £1.5m a year to the party.

Unite has described the confrontation with Progress as part of a "struggle for Labour's soul". Progress, viewed by Unite and others as a "neo-liberal" organisation, is funded by Lord Sainsbury and its president is Stephen Twigg, the shadow education secretary. The GMB, the UK's third largest union behind Unite and Unison, has pledged to "outlaw" Progress, arguing that it has worked to promote its own parliamentary candidates and has pressed the Labour leadership to accept public sector pay restraint.

However, the Labour leader, Ed Miliband, has defended Progress, describing it as a "good organisation," while Lord Mandelson has praised the thinktank as "forward looking" and "modernising". The GMB, meanwhile, has likened it to the militant tendency, a faction that tried to get Labour MPs deselected in the 1980s.

Prentis also declared that public sector pay will be the next "battleground" for unions. Pledging to co-ordinate strike action with other unions over a multi-year pay freeze, he said: "The pay freeze has to be broken."

Unite has 1.3 million members, with most of its membership split between local government and health employees.