The Conservative campaign in Stoke-on-Trent Central could face a legal challenge after a mailshot claimed two of the city’s Labour MPs – both of whom voted for Article 50 – had “voted against” Theresa May’s Brexit plan.

It is understood that both Stoke North’s Ruth Smeeth and Stoke South’s Rob Flello have sought legal advice.

A Tory leaflet in which the Prime Minister addresses voters in the by-election accuses the two MPs of trying to block Brexit in the Commons last week.

Yet while the decision to vote for the Article 50 bill divided Labour, both Smeeth and Flello voted with the government to begin the process of Brexit.

The letter reads:

“Jeremy Corbyn’s Labour Party can’t agree whether they want to take control of our laws and our borders. They seem to be intent on finding new ways of frustrating the process of leaving. Last week, Stoke’s two other Labour MPs and the Labour MP for Newcastle-under-Lyme [Paul Farrelly, who voted against Article 50] all voted against my plan to deliver Brexit.”

In response, Smeeth tweeted that she was "utterly disgusted" that Theresa May had "lied" about her voting record. A Labour source said the MP, a senior figure in Labour’s by-election campaign, has written to the prime minister to demand a public apology and retraction.

The Conservatives maintain the letters are only referring to MPs' support for amendments to the bill.

A similar leaflet was delivered to homes in Copeland, where Labour faces another crucial by-election next Thursday. It also accused neighbouring Labour MPs of voting to block Brexit the day before last Wednesday’s Article 50 vote. Neither John Woodcock nor Sue Hayman, the MPs for the adjacent constituencies of Barrow-in-Furness and Workington, voted against Article 50.

Utterly disgusted that Theresa May has lied to the people of Stoke about my voting record. This behaviour is a disgrace to her office. pic.twitter.com/lREe3qvsZa — Ruth Smeeth MP (@RuthSmeeth) February 16, 2017

The leaflets matter because both constituencies voted decisively for Brexit – Stoke was 70 per cent Leave, and Copeland 62 per cent. The Prime Minister’s uncompromising rhetoric on the EU has featured prominently in the Tories’ campaign material.

With Labour's existential bind on Brexit increasingly apparent, the row provides an early indication of the sort of campaigns Labour MPs of all stripes will face in leave constituencies at the next general election.

@RuthSmeeth & I will be taking legal advice about this lie by PM @Number10gov https://t.co/sVnOKPyutt — Rob Flello (@RobFlelloMP) February 16, 2017

Sue Hayman, the Workington MP and shadow environment secretary, said she was disappointed that the Tories had tried to spin her votes for Labour amendments to the Article 50 bill as attempts to block Brexit. She said: "I'm disappointed that the Conservatives are trying to claim that Cumbria's Labour MPs are blocking Brexit. While I campaigned for a Remain vote last year, I of course accept the democratic result of the country and of my constituency, which voted to Leave the EU.

"That is why I voted for the bill to trigger Article 50 at both second and third readings in the House of Commons. However, nobody voted to leave the EU in order to make our country or economy poorer, which is why last week I voted for Labour amendments to the Bill on a range of issues, including our continuing membership of Euratom, which was supported by the nuclear industry that is so important to my constituency."

Defending the controversial campaign tactic, a Conservative spokesman said both MPs had backed amendments to the Article 50 bill that “would have put restrictions on the Prime Minister’s negotiation position and curbed her ability to negotiate the very best deal for Britain”.

He said: “The Prime Minister has set out a clear plan for Brexit which Jeremy Corbyn's Labour MPs, including the two Labour MPs in Stoke-on-Trent and the Labour MP in Newcastle-under-Lyme, tried to block or delay in Parliament. Only Jack Brereton will back the Prime Minister’s clear plan to deliver a successful Brexit for Stoke-on-Trent and the United Kingdom.”

A Downing Street spokesperson also confirmed that the letters were intended to refer to Labour MPs' votes for amendments to the bill, and not votes against the bill itself.