Downing Street has previously confirmed officials were looking to house asylum seekers in prefabricated buildings on a range of sites, including industrial land.
Cooper, the new foreign secretary, had also previously said her old department was considering housing people in warehouses.
It comes after estimates that more than 1,000 people crossed the English Channel in small boats on Saturday, Mahmood's first full day in the job, underscoring the scale of the challenge facing her at the Home Office.
A record 29,003 people had crossed the Channel in small boats so far in 2025 at the end of last month, according to the latest official figures, up from 21,052 for the same period in 2024.
The Conservatives have blamed the increase on Labour's decision to scrap their plan to deport some asylum seekers to Rwanda - which they failed to get up and running before losing the last election.
Speaking to Kuenssberg, Tory leader Kemi Badenoch said Sir Keir "had no plan in place" to replace the "deterrent" offered by their Rwanda plan.
Reform UK, which says it could deport 600,000 people within five years if it takes power, says it would also house people arriving into the UK illegally in prefabricated, or "modular" detention centres, prior to removal.
The party says it would build a series of new removal centres in "remote parts of the country" but has refused to set out particular locations.
It says the new centres would be "basic but not punitive", containing prefabricated two-person rooms, on-site medical facilities, and canteen catering.
Speaking to Sky News, Zia Yusuf, Reform's head of policy, denied this would include shipping containers, adding: "They're not shipping containers, they're purpose-built modular steel structures".