"We deeply regret what happened and the suffering caused," May told the British parliament, as India prepares to mark the 100th anniversary of the killings.
Jeremy Corbyn, leader of the main opposition Labour Party, called for "a full, clear and unequivocal apology".
UK should say sorry for century-old Amritsar massacre, India says
The April 13, 1919, Jallianwala Bagh massacre, in which British troops opened fire on thousands of unarmed protesters, remains an enduring scar from British colonial rule in India.
Colonial-era records show about 400 people died in the northern city of Amritsar when soldiers opened fire on men, women and children in an enclosed area, but Indian figures put the toll at closer to 1,000.
Former British prime minister David Cameron described it as "deeply shameful" during a visit in 2013 but also stopped short of an apology.
A ceremony was due to take place at the site of the massacre on Saturday.
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