The Labour leader, Ed Miliband, will renew calls on public sector unions not to strike this week by saying industrial action will harm their ability to convince the public of their case.

One of Miliband's frontbenchers also waded in with a warning to the public sector unions not to withdraw their support for the Labour MPs who fail to back the strikes.

Miliband's response to the wave of industrial action is potentially politically combustible, even if some of the unions involved in this week's strikes are in unions not affiliated to the Labour party.

In an interview in the Guardian on Saturday, Miliband said he thought it would be a mistake for the unions to strike this week and his aides insist that remains his position.

However, two shadow cabinet members close to Miliband – Peter Hain, the shadow Welsh secretary, and Sadiq Khan, the shadow justice secretary – stopped short of Miliband's warning and instead urged both sides to avoid a strike.

Speaking on the BBC's Andrew Marr Show, Hain said: "I don't think political leaders, in opposition or in government, should either applaud strikes or condemn strikes."

Khan said: "It is a failure on both sides when there is strike action … it's the last, last thing you do and what I'd like to see over the next three or four days is ministers, trade union leaders, speaking and trying to resolve this dispute."

The Conservative party's chairman, Michael Fallon, called on Miliband to make Hain retract his remarks since they contradicted his condemnation of the strike.

Labour officials argued there was no substantial difference across the frontbench, but the dispute comes at a sensitive time for Miliband as he prepares to tell Labour MPs that he plans to take away their right to elect the shadow cabinet. The Labour leader's proposals are part of reforms unveiled at the weekend that would bring about a "serious transformation" of the Labour party.

In an effort to put the party back in touch with the public, Miliband is also seeking to put the ideas of local communities in front of Labour's national policy-setting body, and open up annual conferences to non-affiliated charities and community organisations.

The precise plans on the future of the union vote – currently 50% – are unlikely to be revealed until the autumn, but one idea is to give members of the national policy forum a proportion of the vote at party conference as a way of melding conference and the policy forum process.

The former Labour party leader Tony Blair also urged the unions to show restraint: "The thing about the trade unions is that they too have got to modernise. I mean, I said this constantly when I was leader and they used to think that meant I was anti-union. I'm not, I'm in favour of strong trade unions, I think it's great.

"But you've got to understand how fast the world is changing. And what you've got to be careful of – particularly with public sector unions – is you don't become 'small c' conservatives. You know the world is – we're going to resist the change, we're going to keep the status quo."

Asked about the public sector pensions strikes, Blair said: "I just think the best thing is for them to engage with the process of change."

The shadow transport minister, John Woodcock, writing on the Labour Uncut website, also challenged the unions not to threaten to withdraw support from Labour MPs that fail to back the strikes.

Woodcock, seen as a party moderniser, wrote: "Through all those arguments we should keep in mind that there never has been a time when the Labour party was completely in line with any one group who supports it – nor will there ever be.

"And whatever views any supporter may have about an individual candidate at a general election, each one stands on a shared platform with an agreed manifesto. So an organisation that wants Labour to win but refuses to support some of the candidates surely risks shooting itself in the foot.

"And, to extend the metaphor, it risks shooting in the foot the millions of working people it represents."