The local authority has effectively suspended its existing rules, including a maximum vehicle turning circle of just 25ft.
Because Allied's wheelchair-friendly vehicles are slightly larger than a London-style cab, this rule had prevented it from supplying Manchester's taxi drivers.
Manchester has changed its approach due to concerns that the collapse into administration of Manganese Bronze, the dominant supplier of London-style cabs, is hindering the supply of spare parts and new vehicles.
Paul Nelson, managing director at Allied Vehicles, told The Herald: "They have realised that the world has changed a little bit."
He said Allied could snap up as much as 40% of the Manchester market, where some 1250 cabs operate.
Mr Nelson said he expected Allied to add 20 to 30 jobs to cater with the upturn in business. It currently employs 370 people.
Nigel Murphy, Manchester City Council's executive member for the environment, said: "A review of our current hackney carriage licence policy will take place over the next six months following the news that Manganese Bronze, the parent company that manufactures the London-style black cabs, has gone into administration.
"Hackney carriages are an important means of getting around Manchester and it's important we get this right and make sure our high standards, particularly regarding disabled access to these vehicles, are maintained."
In the meantime, it has agreed interim measures with a group of council officers and elected members considering applications for taxi licences.
Previously it used the same rules as Transport for London, which included the requirement that taxis complete a turning circle within a space equivalent to that available outside the Savoy hotel on The Strand.
Allied, whose headquarters are 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. Manganese Bronze fell into administration after having to recall 400 vehicles owing to a steering fault.