Labour’s deputy leader, Angela Rayner, has privately expressed her objections to Keir Starmer’s last-minute bid to push through changes to the party’s leadership election rules, the Guardian understands.

Rayner, the shadow secretary of state for the future of work as well as Starmer’s number two, has made clear she would like the party’s Brighton conference to focus on attacking the government and setting out Labour’s offer to the country, rather than wrangling over internal rule changes.

She is understood to be concerned by the timing of the changes and the principle of ditching one-member-one-vote.

Rayner is the latest senior party figure to question Starmer’s tactics, with the battle over the proposals raging just as he prepares to set out what Labour stands for in a keenly awaited party conference speech next Wednesday.

The Scottish Labour leader, Anas Sarwar, added his voice to those questioning the move, telling the Daily Record: “I don’t think it should be our focus. It is certainly not my focus. I’m going to conference to talk about the issues I care about.”

The Labour mayor of London, Sadiq Khan, also appeared lukewarm about the changes this week, saying: “It’s not at the fore of my mind.”

Many senior party figures were blindsided when Starmer announced the shadow cabinet this week that he would like to see a return to an electoral college for choosing his successor, reversing the shift made by Ed Miliband to a one-member-one-vote system.

An electoral college would give unions and Labour MPs one-third of the vote, with members making up the rest. The backing of the unions helped Miliband to clinch the leadership over his brother David in the 2010 contest.

Labour members tend to be more leftwing than MPs, so the move is seen by many on the left as an attempt to avoid a repeat of the 2015 leadership contest, which swept the radical outsider Jeremy Corbyn into the leader’s office.

However, critics claim Starmer – who has previously stressed the importance of party unity – is being drawn into a factional showdown with the left when he should be challenging Boris Johnson over the cost-of-living crisis.

Labour-supporting trades unions will consider the proposals for a second time at a meeting at 5pm on Friday, and talks have been continuing with the leadership. Starmer is understood to be keen to press ahead with the plans in some form.

The party’s national executive committee will then hold talks from 7pm on Friday. If the proposals are adopted, they are likely to be put to conference for approval over the weekend, most likely on Saturday.

Sarwar’s deputy, Jackie Baillie, is proposing an amendment to the electoral college plan in which MSPs, Welsh assembly members and Labour councillors would be given a vote as well as MPs, to better reflect the makeup of the party’s elected representatives.

As well as changing the leadership rules, Starmer also wants to streamline Labour’s policymaking process, shifting power away from the annual conference; and to make it harder to reselect sitting MPs.

Unite’s new leader, Sharon Graham, has criticised the plans, writing to Labour MPs to urge them to reject the return to an electoral college, calling it “unfair, undemocratic and a backwards step”.

Unison is understood to be unlikely to make a firm decision until its executive meets on Saturday morning.

Recent Labour conferences have frequently been marred by battles over rule changes, including a failed attempt in 2019 to abolish the deputy leader Tom Watson’s job on the eve of the annual gathering.