Theresa May has reportedly moved to quash an attempt by her cabinet colleague Sajid Javid to improve mobile phone coverage by warning that the plans could aid terrorists, according a leaked letter.

As David Cameron praised the home secretary as tough and robust, the Times published details from the letter which showed May has warned that the culture secretary’s pet project to end mobile “not-spots” could have a “detrimental impact on law enforcement”.

Javid will launch a consultation designed to force mobile companies to provide a more uniform service across the country. The culture secretary, who unveiled his plans at the Tory conference last month after David Cameron complained of patchy mobile phone reception in Cornwall, will say that in 20% of the UK people cannot make calls.

But May argues in the leaked internal Whitehall letter that Javid’s plans to end “not-spots”, by allowing customers to roam between rival networks, could aid criminals and terrorists. The Times reported that May’s objections centre around concerns that roaming would make it more difficult for the agencies to track suspects.

In the letter, extracts of which have been published in the Times, May says that national roaming “could have a detrimental impact on law enforcement, security and intelligence agency access to communications data and lawful intercept”.

The home secretary, who also reportedly objected to the likes of Tesco offering customers mobile phone packages with access to the four main networks, called for studies to ensure the changes do not prevent police from having access to “information that is crucial to keeping us safe”. The ability of the intelligence agencies to monitor calls is, she wrote, “vital to protecting the public from crime and terrorism”.

The home secretary’s concerns about national roaming will be welcomed by mobile phone companies who are alarmed by Javid’s plans.

The culture secretary has said it is wrong that visitors from other EU member states are automatically switched, under EU law, to a new provider when they lose a mobile phone signal unlike domestic users whose phones do not switch in the same way.

The intervention by May is likely to revive criticism that she often acts in an uncollegiate way, a point made by the Liberal Democrat home office minister Norman Baker when he resigned this week. May’s letter may also be seized on by civil liberties campaigners who say she appears not to challenge the views of the intelligence agencies.

The leaking of the letter came after the prime minister praised May for her robust approach after the resignation of Baker. Speaking during the Rochester and Strood byelection, Cameron said: “I want someone who is a tough home secretary, it’s a tough world out there. You’ve got to deal with crime, you’ve got to deal with terrorism, we’ve needed to reform the police, we’ve needed to take a whole series of actions to keep this country safe. Yes, she is tough and robust but that’s why I think she is so good.”