The Prime Minister intervened after being told the department did not plan to introduce new immigration rules for at least two years following Brexit.
She rejected pleas from senior Whitehall officials that the new system would not be ready in time.
Mrs May wants to ensure that EU migrants who arrive after March 29, 2019 do not get the automatic right to settle permanently in the country – currently available under Brussels’ free movement rules.
News of Mrs May’s decision was being seen last night as confirmation that Downing Street is increasing efforts to secure Britain’s borders for Brexit.
It follows warnings from the Migration Watch think-tank that around a million more EU citizens could get rights to settle in the UK permanently if new border rules are delayed for two years after Britain’s departure.
“The Home Office announced last summer that the system of registration would be introduced for people arriving after March 2019 – that work is well under way,” a Downing Street spokesman said.
“We are confident it will be finished on time.”
He was responding to suggestions from Whitehall sources that work on the new system was lagging behind.
One Government source had said: “That work has barely begun and I don’t think anyone has any confidence that such a new system can be ready for March next year.”
But senior Conservative backbencher Jacob Rees-Mogg, chairman of the 60-strong European Reform Group of Eurosceptic MPs, rejected the claims about Home Secretary Amber Rudd’s department.
He said: “If this were true, it would be a sad admission of incompetence at the Home Office and it would be hard to believe that someone as efficient as Amber Rudd would accept such a sorry state of affairs.”
The Home Office said: “The Prime Minister has been clear that during the implementation period there will be a registration system for EU citizens coming to the UK.
“The precise details of the implementation period are currently being negotiated with the EU, but planning is well under way.”
Mrs May will deliver a speech on post-Brexit UK-EU security soon and will complete the process in three weeks’ time with an address on the overall relationship following a Cabinet withdrawal committee summit.