Ministers held an emergency Cobra meeting on Saturday amid forecasts that the record for the hottest day will be broken.
The Met Office’s highest alert, meaning a risk to life, has been issued in areas including London, Manchester and York.
TfL chief Andy Lord has now warned customers to only travel is their journey is essential.
He told LBC: “We’re advising all our customers to only travel if their journey is essential, to make sure that they stay hydrated and carry water with them if they do have to travel.
“Check before they travel because journey times will be extended. We will have reduced services across the TFL network because of the safety restrictions we need to put in place due to the heat.”
“When the rail temperature gets above 50C, we have to reduce the speed to reduce the risk of damaging the track. In extreme cases, the rails can buckle, we need to avoid that and maintain the safety of the railway.”
Meanwhile, the UK Health Security Agency issued its “national emergency” level-four heat alert to medical and care organisations.
It said illness and death could occur among the “fit and healthy”.
“On Tuesday July 19, DO NOT TRAVEL between #LondonKingsCross and south of #York & #Leeds, as no trains will run,” it said in a tweet.
“Only travel if necessary to other destinations.”
People who have already bought tickets are urged to visit the LNER website for information about refunds.
The chairman of the NHS Confederation has said hospitals are going to be “really, really pushed” over the next few days.
Lord Victor Adebowale said the NHS “will cope” over the next few days but added “coping isn’t good enough”.
He told Times Radio: “My members are pretty stretched at the moment. Ambulances are operating at their peak, the waiting times for ambulances are now getting longer.
“We are going to be really, really pushed and it’s not just the red warning, the heatwave. We are dealing with Covid, which is causing sickness in our ambulance crews, which is one of the reasons we’ve got vacancies. And we’ve also got a social care challenge, which is we can’t get people out of hospital because social care is on its knees.”
Lord Adebowale added: “The NHS will cope but coping isn’t good enough. We need to be actually operating in a way that allows people to get the treatment they need wherever needed, in good time. And that’s a struggle.”
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