Dutt, who has already undergone imprisonment for 18 months, was sentenced on Thursday to five years' imprisonment in the Mumbai serial blasts case.
Talking to reporters after the meeting at Raj Bhawan this evening, Singh sought mercy for the actor saying, "Justice delayed is justice denied. Justice hurried is justice buried".
"Has Sanjay Dutt committed any terror activity in the last 20 years? Justice delayed is justice denied," Singh said.
"The Governor and President can use their powers to pardon Dutt," Singh said, after meeting the Governor.
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"I hope that you will be kind enough to exercise your discretionary powers to grant amnesty to Sanjay Dutt, considering his background, contribution to cinema and his good conduct in the past 20 years," she wrote.
She also attached a letter from the Sewagram Ashram in Wardha, written last year to former President Pratibha Patil.
"That letter has appreciated Dutt's contribution to Gandhism which has influenced the young generation," she said.
Singh has been close to Dutt and was his mentor when the actor briefly dabbled in politics.
"He (Dutt) has already been punished enough. Twenty years is almost like life imprisonment. I mean, he suffered for twenty years and I think to do this to him at this time...I am going to personally meet the Governor," Bachchan had told reporters outside Parliament House recently.
Sinha had joined Justice Katju in appealing to the Maharashtra Governor to pardon Dutt after the Supreme Court upheld his conviction in the 1993 Mumbai blasts case.