Competing claims about levels of voter registration in Scotland have cast doubt on the idea that a "missing million" of previously unregistered voters could swing the referendum result in favour of independence.

On Monday, Scotland's first minister, Alex Salmond warned unregistered voters that they had "just 24 hours left to sign up to take part in Scotland's date with destiny". Voters have until midnight on Tuesday to register for the independence referendum on 18 September.

Salmond estimated at the weekend that 180,000 extra voters had now signed up. But Scottish government officials said there was no official source for the first minister's figure; that was based on an extrapolation, they said, from new-voter figures released by some councils. Even so, there was clear evidence that there had been "a huge increase in people signing up to vote".

The phrase "missing million" is rather loose, and can be taken to mean those who are registered but habitually do not vote, as well as those missing from the register entirely.

According to the general register of Scotland, the population aged 16 and over is about 4.475 million. This includes the additional 124,000 16- and 17-year-olds now able to register following the extension of the franchise to them for the referendum.

The number of voters signed up, as of the start of August, was 4,166,683, indicating that only 300,000 potential voters remained off the register. But whether significant numbers of these people had registered over the past month, and whether a critical mass of those would be voting yes, was in dispute.

One source of data for the Scottish government's figure was the disclosure by City of Edinburgh council officials that the number of new voters in the Lothians, an area covering the four councils of West Lothian, East Lothian, Midlothian and Edinburgh, had risen by 31,600 people, about 4.9%, over the last three months.

The Edinburgh Evening News reported that the total Lothians electorate had now hit 645,000, against an estimated 705,000 people eligible to vote.

"I think that's probably as many as you could hope to get for Edinburgh and the highest we've ever had," said Joan Hewton, Lothian's electoral registration officer.

Scottish Labour sources were sceptical about Salmond's estimates. Labour and Better Together campaigners claimed they had brought in thousands of new voters, who would back the union.

Labour said Edinburgh had shown a spike in registrations last month The highest figure was in Edinburgh Central, where 1,172 new voters registered. 1,104 extra voters registered in Edinburgh North and Leith – a key target area for the Radical Independence Campaign (Ric), which has been registering new voters in working-class areas across the country.

However in other urban centres that are a focus for the Ric and Yes Scotland, such as Dundee and urban Glasgow, the new voter figures for August were much lower. They totalled 384 in Dundee, and for Glasgow reached just 39 in Provan, 50 in Pollok and 27 in Shettleston.

They said that some of those voters could easily be accounted for by the normal circulation of people moving into areas, such as students or 16-year-olds registering for the first time, rather than representing unregistered adults.

"Broadly speaking, this hasn't shifted at all since July. From our own work, speaking to undecided postal voters across all ages, all demographics, they break more to no [votes] than to yes," one source claimed.

But figures obtained by the BBC's Scotland 2014 programme showed that in the past seven days Glasgow had received more than 9,000 applications, Renfrewshire 5,000, and Shetland and Orkney almost 1,000.

Edinburgh has received 30,000 applications in the past two months. Officials said these numbers had yet to appear on official lists because of backlogs caused by the unprecedented numbers, but insisted they would meet the deadline of processing them by 11 September.

Referring to the "missing million," Jonathan Shafi of Ric said he believed many of this key cohort were already registered to vote but from habit had not: "These are people who are not apathetic but are alienated from the political process."

He said that at a registration drive in Glasgow, on Saturday, 300 people had signed to vote within the space of three hours, filling out forms at street stalls that were later delivered to the authorities by Ric activists.

Shafi said he could not be certain that all the people registering were not already on the voters' roll, but insisted the majority would be voting yes.

Shafi added that it was impossible to know exactly how many new voters had been registered throughout their campaign, but estimated that about 1,500 had signed up across Scotland in the past week alone.

The Electoral Commission said on Tuesday that 100,000 registration forms had been downloaded from its website since it launched its voter registration drive three weeks ago.