The Indian foreign ministry tweeted that “no such request” had been made, adding that all issues with Pakistan were “discussed only bilaterally”.
Mr Trump’s claim that he had been asked came as he hosted Pakistan’s Prime Minister Imran Khan at the White House.
Both India and Pakistan claim all of Kashmir, but control only parts of it.
Indian Foreign Minister S Jaishankar repeated in the parliament that Mr Trump’s claims were not true.
But opposition MPs have demanded that Mr Modi should address the parliament on this issue. Rahul Gandhi, who recently resigned as the chief of main opposition Congress party, tweeted that the foreign ministry’s denial wasn’t enough.
President Trump says PM Modi asked him to mediate between India & Pakistan on Kashmir!
If true, PM Modi has betrayed India’s interests & 1972 Shimla Agreement.
A weak Foreign Ministry denial won’t do. PM must tell the nation what transpired in the meeting between him & @POTUS
— Rahul Gandhi (@RahulGandhi) July 23, 2019
The neighbours have fought two wars over Kashmir, and tensions flared again in the Muslim-majority territory in February leading to cross-border air strikes.
Mr Trump had his first face-to-face meeting with Mr Khan on Monday at the Oval Office.
Afterwards they spoke to reporters and addressed a variety of subjects.
When PM Imran Khan was asked if the US could help in the 70-year-long dispute between India and Pakistan in Kashmir, he said: “Only the most powerful state headed by President Trump can bring the two countries together,” according to a White House transcript.
Mr Trump then added: “So I was with Prime Minister Modi two weeks ago and we talked about this subject and he actually said, ‘Would you like to be a mediator, or arbitrator?’
“I said ‘where?’ He said: ‘Kashmir, because this has been going on for many, many years.’
“If I can help, I would love to be a mediator,” Mr Trump said.
Pakistan welcomes mediation by a third party in Kashmir, while India says all issues should only be discussed bilaterally.
India was quick to respond to Mr Trump’s statement.
“We have seen [Mr Trump’s] remarks to the press. No such request has been made,” said Raveesh Kumar, India’s foreign ministry spokesman, in a tweet on Tuesday.
“It has been India’s consistent position that all outstanding issues with Pakistan are discussed only bilaterally.
“Any engagement with Pakistan would require an end to cross-border terrorism.”
Indian politician Shashi Tharoor criticised Mr Trump’s remarks, saying he did not have the “slightest idea of what he was talking about”.
“He has either not been briefed or not understood what Modi was saying,” he said on Twitter.
A military stand-off in February ratcheted up tensions between the neighbours over the region again when India ordered a pre-emptive strike on what it said was a terrorist training camp in Pakistan.