It is the latest blow for workers at the oven manufacturer, which closed its historic Coalbrookdale foundry last year and made nine people redundant last month.
The jobs, believed to be engineers, will go if Aga goes ahead with outsourcing some of the work currently done in-house.
A 30-day consultation involving workplace representatives is under way.
It is not known how many of the latest job losses, likely to be made in the summer, will be at the Ketley headquarters.
The company could only say that potential redundancies had been identified across the UK.
A statement from the company said: "Consideration is currently being given to the possibility of outsourcing some work and, as a result of this, we are about to enter into consultation with workplace representatives about potential redundancies.
"This could affect up to 39 staff currently employed throughout the UK and is envisaged to last 30 days.
"We will, of course, do everything we can to ensure the process is managed with sensitivity.”
The firm's branch in Ketley employs about 200 people.
Stuart Baker, regional officer for union Unite, said he was extremely disappointed that Aga had chosen not to consult with the trade union.
Invented in Sweden, Agas had been made in Shropshire since the 1930s and their popularity with the middle classes and rural families made them legendary.
But in 2015 Aga Rangemaster, as the company was called, was sold to Illinois-based food services giant Middlebuy Corporation for £129 million.
The Aga foundry shut for the final time at the end of November, with the 42 remaining workers leaving their boots and helmets on the famous "Coalbrookdale Co” gates as they left. Some were offered jobs at a foundry in Walsall run by H Goodwin.