East coast main line operator LNER is imposing the restriction to ensure passengers can observe minimum distancing aboard.
It operates Edinburgh-London services, some of which continue between the Scottish capital and Aberdeen, Inverness and Glasgow.
LNER’s services will increase on Monday from about 35 per cent of normal levels to 60 per cent.
A spokesperson said: “To support physical distancing for customers who do need to travel with us at this time, all LNER services will be reservation only from Monday 18 May.
“We will have full details updated on our website over the weekend.
“LNER services continue to operate for those who have no alternative travel options.”
Avanti West Coast, which trains between Glasgow, Edinburgh and London via the west coast main line, said the capacity of its trains will be limited to around a quarter of normal levels.
This means passengers who have bought a ticket without a reservation may not be allowed to travel on the next train that arrives.
The measure is being introduced ahead of timetables being increased on Monday and following updated travel advice by the UK Government.
Phil Whittingham, managing director of Avanti West Coast, said: "We're appealing to our valued customers to help us and other passengers by only travelling with a reservation.
"If everyone does this, we'll be able to keep social distancing in place on board, both for our customers and our people.
"If customers do turn up without a reservation, we'll do our best to help but we can't guarantee they'll be able to take the train they want."
The firm has introduced a series of additional hygiene measures, including issuing staff with face masks, enhanced cleaning regimes and closing waiting rooms and lounges.
Services run by English-based train operators will be ramped up from around 50 per cent of the normal timetable to 70 per cent from Monday as part of the easing of coronavirus lockdown restrictions south of the Border.
But a planned 4 per cent increase at ScotRail – currently running at 43 per cent of normal – has been shelved because lockdown restrictions have not been eased in Scotland, and after lobbying from the rail unions who said boosting services would compromise the Stay at Home message.
Industry body the Rail Delivery Group (RDG) is urging passengers to avoid non-essential travel after recording an increase in journey planning of nearly a quarter this week.
It said 3.7 million journeys were planned between Sunday and Thursday, compared with three million over the same period one week earlier.
The RDG warned that the capacity of trains will be restricted to as little as 10 per cent of the normal number of people who travel.
Those who have no alternative means of making their journey are asked to avoid peak times.
RDG director of nations and regions Robert Nisbet said: "To keep people safe, there will only be space on trains for as few as a tenth of the usual number of people despite train companies gradually increasing services next week.
"We need people's help to keep trains clear for those who really need them, so we are asking people to consider alternatives like cycling or walking, and if that's not possible, to travel at quieter times.
"We are doing everything possible to keep passengers and staff safe, including cleaning trains and stations several times a day, managing capacity and increasing signage to help people get around.
"For those who do need to take the train, we ask that they plan ahead, consider others and stay safe when they use public transport.
"That means buying tickets online for quieter times of day, maintaining social distancing wherever possible, wearing a face covering and keeping hands sanitised."
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