The head of Britain’s biggest union has made a surprise offer to David Cameron to abandon its opposition to a key part of the crackdown on union rights outlined in the trade union bill – if ministers allow modern and secure online voting on public service strikes.

The olive branch was held out in a letter to the prime minister by Len McCluskey, the leftwing general secretary of Unite, who says “no one can be happy” with staff walking out – particularly in public services such as the railways or the NHS – after low turnouts in strike ballots.

The unexpected move by the man regarded as the most hardline union leader reflects his concern that strikes can be portrayed as illegitimate and irresponsible because the current system of balloting by post brings too low a level of response. Earlier last week the Tories announced the launch of their own trade union movement for those unhappy with leftwing union bosses, prompting fears within Unite about a loss of membership.

In his letter to Cameron, seen by the Observer, McCluskey strikes a conciliatory note, saying he would be prepared to agree to plans for a new double threshold system that would have to be met before a public service strike, if the balloting rules were modernised. He added to the commitment on the Andrew Marr show on Sunday morning.

His stance could him put at odds with Labour leader Jeremy Corbyn and fellow union leaders, who have made clear their strong opposition to the proposed thresholds and to the rest of the trade union bill, which the TUC has called Dickensian.

One of Corbyn’s first acts as leader was to rally Labour opposition to the bill, which proposes a 50% turnout threshold for strikes. It also says that, in key services such as fire, education, health and transport, 40% of eligible voters would have to vote in favour of a strike for it to be legal.

McCluskey, however, suggests a deal that would require an amendment to the bill to bring in secure workplace ballots using online and electronic voting. This system is already used when workforces are asked whether they want to be represented by a union.

“No one, of course, can be happy when strike action takes place – especially in services on which the public depend – on the basis of the active endorsement of only a minority of union members affected,” McCluskey says. “In my long experience of industrial relations, mainly in the private sector, such strikes are a rarity.”

He adds: “It is my understanding that the Electoral Reform Society, which acts as independent scrutineer to most industrial action ballots today, believes there is no difficulty in guaranteeing workplace balloting procedures which are secret and are secure against any possibility of fraud or intimidation.

“Were you to be able to accept this modern and democratic proposal to update balloting procedures then Unite, for its part, would be comfortable about accepting the thresholds and the time limit on the validity of ballots proposed in the trade union bill, without prejudice to our position on other elements of the legislation.”

Sources close to Corbyn expressed surprise at McCluskey’s intervention, stressing that Labour remained opposed to all the main measures in the bill. Others in the union movement said McCluskey may be concerned about being seen to oppose plans with wide public support.