Jilani, who had earlier served in the Pakistan High Commission in New Delhi, said Sharif had "genuine interest" in improving relations with India. (Agencies)
"They will lose a big opportunity if they do not," Jilani, the former foreign secretary, said.
Speaking on Afghanistan, he said the complete withdrawal of American troops from the war-torn country is not desirable.
"Even the talk of US pullout has started having its impact. Pakistan has started to receive more Afghan refugees than before," he said. "This shows that the people of Afghanistan too have fears."
The US plans to withdraw most of its combat troops from Afghanistan by December 2014, although it intends to leave behind a smaller force to help the Afghan government.
The Pak-US relationship will enter a critical phase this year. The Pakistani security establishment fears that India will fill up the vacuum created by the withdrawal.
Pakistan claims India is using Afghan territory for fomenting unrest in Balochistan, a charge denied by New Delhi.
On the other hand, India is wary of Pakistan's role in post-2014 Afghanistan, especially the military's links with militant groups like the Haqqani network, as it believes this will give Islamabad impetus for re-starting the Kashmir jihad.
Jilani, who reached Washington on Saturday, said his first priority would be strengthening trade and economic ties with the world's economic superpower.
"Defence and security ties obviously are equally important but cooperation in the energy sector would be the main task that one has to carry forward."
He agreed with the suggestion that 2014 would be crucial for determining America's role in South Asia, but said it's still early to speculate how the situation would shape up.
"In case the bulk of the troops withdraw, the major responsibility would lie on our shoulders. Security responsibilities, previously being shared by others, then exclusively become Pakistan’s responsibility. It is certainly going to be a big challenge," he said.
Commenting on a recent US intelligence report which predicted that Taliban would regain their influence if NATO forces withdrew, Jilani said Taliban were an important factor in Afghanistan and so far all efforts to bring about reconciliation between Taliban and others produced no results.
"From our assessment, they will continue to play a very important role and that's why Pakistan feels that this process of reconciliation is key to stability in Afghanistan," he said.
Reconciliations between the Taliban and the Afghan government and among all other factions were both extremely important, he added.
Jilani, who had earlier served in the Pakistan High Commission in New Delhi, said Sharif had "genuine interest" in improving relations with India.