Allied's managing director Paul Nelson believes access to the London market could boost its sales by 500 vehicles a year and enable it to create 100 jobs.
Mr Nelson said: "We have just put an offer into Transport for London and the Mayor's office in London. We have offered to loan vehicles and build new cabs if they relax the conditions of fitness."
He added: "We have a vehicle going down for testing next week."
Mr Nelson wants to be allowed to supply Allied's Peugeot E7 taxi – which was revamped earlier this year – to the London licensed trade.
However, the wheelchair-friendly vehicle falls foul of the stipulation that only a taxi that can turn within 25 feet is eligible to be licensed as a black cab.
This matches the space available for cabs to turn outside the Savoy hotel on The Strand. Transport for London has hitherto resisted changing the rules.
Mr Nelson said: "It is not a safety issue, it is a nicety.
"We are a British company. But we cannot sell into our own capital city."
The London taxi market was solely supplied by Manganese Bronze until 2008 but now Mercedes Benz has 40% of sales. Mercedes could be left as the sole supplier if administrators at PricewaterhouseCoopers fail to find a buyer for Manganese Bronze.
Licensed Taxi Drivers' Association general secretary Steve McNamara said: "We are concerned there is only one vehicle supplier at the moment and we do not think a monopoly supplier is good for the taxi trade, taxi drivers or for the public."
Manganese Bronze, which had been making cabs since 1948, fell into administration after having to recall 400 TX4 black cabs owing to faulty steering boxes.
This has left drivers struggling to find replacement parts and new vehicles.
Exacerbating the situation is the fact that cabs 15 years old or more were withdrawn from London's streets last year in a clear air campaign.
Part-time taxi drivers many of whom rent cars during the festive season now face long waiting lists.
Allied, which is headquartered in Possil in the north of Glasgow, has already lent 20 taxis to drivers in Edinburgh and Glasgow whose Manganese Bronze vehicles have been taken off the road. Established by ex-mechanic Gerry Facenna and his brother Michael in 1993, it specialises in vehicles for use by disabled people.
Mr Nelson argued wheelchair users in London would welcome more spacious cabs that do not force them to sit sideways.
Mr Nelson believes Allied could take one-quarter of the London taxi market, or some 500 vehicles a year. It currently sells 4500 vehicles annually. This could add another 100 workers to its current 370-strong payroll.
The development comes after what Mr Nelson described as a "solid" last financial year. Accounts for the year to April 30, 2012, show turnover rose 6.8% to £65.7 million. Pre-tax earnings were up 2.3% at £744,000.
A Transport for London spokesman said: "Allied Vehicles have formally applied to licence one of its vehicles from elsewhere in the UK as a London taxi.
"We are progressing this application, which will include a rigorous inspection to ensure the vehicle meets all the licensing requirements of a London taxi.
"The inspection process will begin later this month."