The former Bond girl has been involved in early workshops for the production from a script by One Man, Two Guvnors writer Richard Bean.
It is understood Arterton would be in line to play the lead role of Rita, played in the film by Sally Hawkins.
Producers Stage Entertainment told the BBC they would "very much love her to be part of the production".
It has not yet been announced which London theatre the musical will be staged at, but it is hoped it will play next year.
"As with all casting there are schedules to work out, but we'd very much love [Arterton] to be part of the production when it opens in the West End," a spokesman said.
Early workshops were directed by Rupert Goold - the new artistic director of London's Almeida Theatre - with the score written by David Arnold, who also provided the music for the Bafta-nominated film.
Made in Dagenham told the story of the machinists who went on strike in 1968 at the Ford Dagenham car factory demanding equal pay and protesting against sexual discrimination.
The character of Rita played a pivotal role in leading the strike and meeting former employment secretary Barbara Castle, leading to the Equal Pay Act 1970.
The film also starred Andrea Riseborough, Jaime Winstone and Bob Hoskins.

The FBI seized computers and mobile phones from the home in October, after celebrities including Jennifer Lawrence had pictures stolen - apparently via a hack on their Apple iCloud accounts.
Court papers have identified eight celebrities by their initials only.
The FBI said no charges have been filed at the current time.
Last summer's scandal highlighted the vulnerabilities of cloud computing - which allows personal information to be stored on the Internet and accessed from any device.
The affidavit - a sworn statement to the court - has identified eight celebrities by their initials only: AS, CH, HS, JM, OW, AK, EB and AH.
One of the identified stars learned she was a victim of the leak through the news media. The papers have stated that "all photos were taken with her iPhone and sent through iMessage to her boyfriend".
Actress Jennifer Lawrence has since spoken about the leak. Speaking to Vanity Fair, the 24-year-old said: "It is not a scandal. It is a sex crime. It is a sexual violation. It's disgusting."

The full-back for Pro12 side Scarlets appeared with four team-mates in fancy dress where he posed as Wilfried Bony.
Mr Williams was criticised by Show Racism the Red Card, which said it was "disappointed to see the image".
The photo has been removed and he said he was not a racist and there was "no malice behind it".
A statement from the Scarlets said: "Scarlets is fully committed to eradicating discrimination and promoting equality for both staff and our customers by treating people fairly and according to their needs.
"Scarlets is proud of its record in promoting equality and non-discriminatory behaviour amongst all our employees and this applies to the way we behave to members of the public and to other employees."
Gemma Arterton is set to star in a West End musical version of the 2010 film Made in Dagenham.
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Newly-released legal documents have revealed a building in Chicago was raided after hundreds of nude photos of female stars were leaked last year.
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Wales rugby union international Liam Williams has apologised after posting a picture on Twitter of himself "blacked-up" as a Swansea City footballer.