The Labour leadership contender Liz Kendall has reacted furiously to claims by Lord Falconer, a prominent supporter of her rival Andy Burnham, that the two women in the party’s leadership race are not up to the challenge of the job.

Kendall believes the comments from Charles Falconer, the shadow lord chancellor, about her and Yvette Cooper – reported in the Times under the headline “Women not tough enough to lead Labour’’ – had sexist undertones.

Falconer was quoted as saying: “Neither Yvette nor Liz can steer the Labour party through the challenging few years ahead when we need a leader who can reach out to all wings of our party and provide unity. As a result, both Liz and Yvette are unlikely to beat Jeremy [Corbyn, the fourth leadership candidate].”

Kendall hit back on Friday: “It is depressing to see a senior man in the party dismiss the contribution of women so easily.

“For Charlie to say that women somehow aren’t tough enough to lead the Labour party is a gross insult and, as for standing up to Jeremy Corbyn, I’m the only candidate who has been saying he would be a disaster for our party and that I wouldn’t serve in his shadow cabinet, unlike the candidate Charlie is supporting.”

Kendall added: “Charlie made a great contribution to the last Labour government and I would have thought he would have learned that one of the reasons we achieved so much was because there was a record number of women around the top table.

“That’s what delivered policies such as the first national childcare strategy that I was proud to be part of developing. Now it’s time to elect [Labour’s] first woman leader and first woman prime minister, so that we can do great things again, such as deliver a real living wage, tackle the inequality in early years and ensure dignity for older people by reforming social care.”

A spokesperson for Andy Burnham said: “The quotes attributed to Charlie Falconer in the headline of the Times are not something he said or believes. People should read Charlie’s article to see he doesn’t even raise the question of whether anyone is ‘tough enough’ or anyone’s gender.”

The latest tally of constituency Labour party nominations showed Cooper gaining ground with a haul of 14 CLPs last night. The latest numbers showed Corbyn on 91, Burnham on 83, Cooper on 79 and Kendall on 14.

In the equivalent campaign in 2010, the leftwing candidate Diane Abbott won only 20 nominations, underlining the extent to which the party’s mood, and possibly membership, has changed.

A nomination by a CLP gives no guarantee that the bulk of the membership in a constituency will follow suit, and much will depend on the turnout of more established members who may think Labour’s re-election lies in going further left.

The Burnham camp point to a new Standard/Mori poll showing their man as the public’s (not just the party’s) choice as the best prime minister among the leadership rivals. In a sign of how the party is struggling, Burnham is the only candidate without a negative rating with the public.