The UK Foreign Office statement comes after Mr Khan during his visit to Amritsar yesterday said the British government should apologise for the mass killing.
"I'm clear that the government should now apologise, especially as we reach the centenary of the massacre. This is about properly acknowledging what happened here and giving the people of Amritsar and India the closure they need through a formal apology," Pakistani-origin Mr Khan said during his ongoing trade mission to India and Pakistan.
He described the massacre as one of the most horrific events in India's history.
The Foreign Office invoked former British Prime Minister David Cameron's views on the issue after Mr Khan asked for an apology.
"As the former Prime Minister said when he visited the Jallianwala Bagh in 2013, the massacre was a deeply shameful act in British history and one that we should never forget. It is right that we pay respect to those who lost their lives and remember what happened. The British Government rightly condemned the events at the time," it said in a statement.
The UK's Conservative party-led government had last fallen short of a formal apology for the massacre during a visit to Amritsar by Mr Cameron.
During his Indian trade mission in February 2013, Mr Cameron had said it would be wrong to "reach back into history" and apologise for the wrongs of British colonialism.
Virendra Sharma, a fellow Opposition Labour Party member like Mr Khan, had launched the petition on the UK Parliament's website earlier this year but it has attracted just over 1,778 signatures.
At 10,000 signatures, the UK government would have to respond to the petition and at 100,000 it has to be considered for a House of Commons debate.
"In 1919 Colonel Dyer ordered his men to fire, and maybe 1,000 peaceful protesters were left dead. At the time Winston Churchill proclaimed the massacre 'monstrous' and the British Government condemned Dyer for his actions, but no apology has since been forthcoming. It is now time to apologise," the petition reads.
Punjabi-origin Mr Sharma has been campaigning for an apology over the years.
Mr Khan, who was on a three-city tour to India, visited New Delhi, Mumbai and Amritsar to strengthen the UK's capital trade ties with India.