The order will not apply to a further three million customers in the Thames Valley where rainfall has been higher, the company said. But Thames Water has not ruled out applying for a drought order extension to include that area later in the year.
If granted, the drought order will be England's fourth this year following Sutton and East Surrey Water, Southern Water and Mid Kent Water which already have them. It is the first application for such an order in London since 1976.
Thames Water said the restrictions would be needed in the event of a hot, dry summer but not if "normal" levels of summer rainfall occur. The application is likely to outrage Thames Water's critics who say the company loses too much water through leakages.
Labour's Diane Abbott, MP for Hackney North and Stoke Newington, accused Thames Water last month of making consumers pay for its "failings" - and of losing 915 million litres of water per day.
If introduced, the restrictions could ban the use of hosepipes and sprinklers to water public gardens, parks and golf courses. Filling private swimming pools, washing the outside of buildings and washing windows with hosepipes or sprinklers could also be stopped.
Jeremy Pelczer, chief executive officer of Thames Water, said the company could need to implement the drought order in London by early August. "May's above-average rainfall dampened demand, boosted river flows and allowed us to recharge reservoirs, but the underlying situation remains serious with groundwater levels in most areas very low," he said.
"We simply don't know how much rain we will get in the rest of the summer, and if the weather is unusually hot and dry, we will need to restrict water use more than we are at present. So we believe it is sensible to apply for a drought order now, even though normal levels of summer rainfall would mean that further restrictions are not needed."
Applying for the order is a "essential step" in gaining Environment Agency permission to take more water than usual from rivers to help maintain reservoir levels, Mr Pelczer added.
Thames Water said a further application for an "emergency" drought order to permit water rationing and standpipes in the streets was "a remote possibility this year". Two dry winters in a row have caused "very low" base river and groundwater levels across the Thames Water region, the company said.
Its drought order application will be subject to an independent inquiry before Defra decides whether to approve it. Mr Pelczer said Thames Water was carrying out a £1 billion programme to bring down London's "unacceptably high leakage rates". The company brought in a hosepipe ban on April 3 this year - its first in 15 years.
Consumer Council for Water spokesman Andrew Marsh said: "I am very concerned because Thames Water have a poor record on containing leakages from their network and so they will find it harder than other companies to take customers with them in conserving water resources.
"Our particular concern is about those businesses which are water-dependent because their livelihoods could be affected by a drought order. The Government needs to think about all those businesses when deciding whether to grant the order."
Alan Bradley, Westminster City Council Cabinet member for the Environment, said: "It's pretty galling at a time when Thames Water loses 900 million litres a day for them to ask for a drought order to be imposed on Londoners."