She said it showed Britain’s “steadfast commitment” to the alliance and how it can rely on the UK.
The extra personnel requested by Nato will take the UK’s commitment in Afghanistan to around 1,100.
They are part of Nato’s non-combat Resolute Support mission to build up the capacity of Afghan defence and security forces to combat Taliban and other terror groups and work to create the conditions for peace talks.
The extra troops will boost the British-led Kabul Security Force, which plays a critical role in enabling Nato advisors to do their work, and they will also mentor Afghan forces who are responsible for security in the city.
President Trump is expected in Brussels again to berate Nato members who – unlike Britain – do not meet the alliance target of spending at least 2 per cent of GDP on defence, and some fear he will order scaling-down of American military involvement in Europe.
Mrs May said: “Nato is as vital today as it ever has been and our commitment to it remains steadfast.
“The alliance can rely on the UK to lead by example, not just in meeting the 2 per cent pledge but by contributing our cutting edge capabilities to operations around the world.
“In committing additional troops to the Train Advise Assist operation in Afghanistan, we have underlined once again that when Nato calls, the UK is among the first to answer.
The extra troops will initially come from the Welsh Guards, with around half going next month and the rest by next February.
Afghanistan is due to hold national elections in October.
Looking ahead to the wider Nato summit agenda, which will include an update on countries’ progress on their pledge at a summit in Wales in 2014 to raise defence spending, Mrs May added: “Playing a leading role in Nato gives us a powerful voice and we use that to repeatedly make the case for a more modern and adaptable alliance.
“At the summit we expect to agree an ambitious package of measures to strengthen our deterrence and defence, increase our efforts on tackling terrorism and address the threats presented by cyber and hybrid warfare.
“And we will continue to make the case for all allies to follow the UK’s example and meet the 2 per cent pledge.”