Jeremy Corbyn has taken another major step forward in his campaign to become Labour leader after the Communication Workers Union announced that it is to endorse him to act as the antidote to the Blairite “virus” in the party. In a sign that the veteran leftwinger is building up huge momentum in his campaign, the CWU general secretary, Dave Ward, announced that he is to follow in the steps of the Unison and Unite unions to endorse Corbyn.

The transport and travel union, the TSSA, also announced that it would back Corbyn on the grounds that he would campaign to “end the austerity quagmire” and that he has backed its calls to take the railways back into public ownership. The TSSA backed Angela Eagle as deputy leader.

The decision of the CWU and the TSSA to follow the two largest – Unite and Unison – in supporting Corbyn is fuelling a growing sense in the Labour party that the MP for Islington North is building a momentum that may be unstoppable. On the eve of the close of nominations by constituency Labour parties (CLPs), the New Statesman has reported that Corbyn is leading the field. It says he has so far secured 133 CLP nominations, followed by 106 for Andy Burnham, 100 for Yvette Cooper and 15 for Liz Kendall.

The CLP nominations carry no formal weight, but they can give a strong indication of how the Labour membership will vote. In 2010, David Miliband won the highest number of CLP nominations and won the membership section of the electoral college. His brother, Ed, overtook him after winning strong union support in the trade union and affiliated societies section of the electoral college. Under the new rules, party members, registered supporters and registered trade unions all have one vote.

In a statement on Twitter accompanied by a YouTube video, the CWU general secretary, Dave Ward, said his union had decided to back Corbyn because it rejected the idea that Labour should occupy the political centre ground. The CWU advised its members to give their second-preference votes to Andy Burnham and to vote for Angela Eagle and Caroline Flint in first and second places in the deputy leadership contest.

Ward said: “I am delighted to announce that the CWU will be backing Jeremy Corbyn MP to be the next leader of the Labour party. There are no quick fixes for the Labour party, but there are some easy decisions, and choosing Jeremy as its leader should be one of them.

“We think that it is time for a change for Labour. The grip of the Blairites and individuals like Peter Mandelson must now be loosened once and for all. There is a virus within the Labour party, and Jeremy Corbyn is the antidote.

“We reject the notion that Labour needs to move to the centre ground of British politics. The centre ground has moved significantly to the right in recent years. We do not see arguing for fairer wealth distribution, decent jobs with good pay, terms and conditions and a substantial increase in affordable housing for the next generation as a leftwing agenda.

“Jeremy agrees with the vast amount of CWU policy, ranging from opposing the selling-off of Royal Mail to offering a real alternative to austerity and repealing anti-trade union laws.”

The CWU has 200,000 members, making it Britain’s fifth biggest union behind Unite, Unison, the GMB – which is not backing any candidate – and Usdaw, which is backing Burnham.

Tom Watson, the former deputy chair of the Labour party, warned that Corbyn may struggle to enforce discipline in light of his record as a rebel in the House of Commons against successive leaderships. Watson told the Huffington Post: “It would certainly be a difficult thing for him to try and command discipline within parts of the parliamentary Labour party with that voting record, but, you know, members are aware of this. It’s their choice.”

Stella Creasy, another candidate for the deputy leadership, criticised what she described as “machine politics” in the Labour movement. This was seen as a swipe at the Labour First group, which is calling on Liz Kendall to ask her supporters to give their second-preference votes to Andy Burnham and Yvette Cooper to stop Corbyn. Her remarks were also aimed at the Unite union, which is supporting Corbyn.

Creasy said: “There’s another side to machine politics, which I also think is turning people off. A way of thinking that has come to do us more harm than good. The sense that the only way to win is through strict uniformity and unquestioning control. A perception then used to justify stitch-ups, petty cliques, back-room deals, and selection fixes. And that feeds the hostility to those in our movement with whom we maybe don’t agree all the time or who are not members at all. It means for too long, factional battles have come before party interest.”