Mr Bambawale had sought the Friday afternoon meeting with Ms Janjua, days after Home Minister Rajnath Singh told parliament that the government would do everything it can to bring justice to Kulbhushan Jadhav.
Sources told NDTV that an appeal against the death sentence was part of the plan that New Delhi is drawing up. Mr Bambawale also conveyed this to Pakistan's foreign office and as the first step, demanded a copy of the charge-sheet and the military court's judgement that sentenced Kulbhushan Jadhav to death.
Pakistan's Defence Minister Khawaja Mohammad Asif told the senate this week that Kulbhushan Jadhav could appeal against his conviction within 60 days, and file mercy pleas to the Army Chief and the President.
But Islamabad has not granted New Delhi consular access to Jadhav despite 13 requests being made over the last year, an average of one request every month. At Friday's meeting, Mr Bambawale made the request again, for the 14th time.
In a statement, Pakistan Prime Minister's Advisor on Foreign Affairs Sartaj Aziz later blamed India for aggravating the situation with its reaction to the verdict. Mr Aziz, who had told the senate in December that the evidence against Mr Jadhav was not conclusive, said his conviction was based on "credible, specific evidence".
New Delhi had reacted angrily, declaring that if the "indefensible" and "farcical" verdict was carried out, India will regard it as a case of "premeditated murder". India also recalled how, shortly before the verdict came, Mr Aziz had told the senate that the evidence against Mr Jadhav was not conclusive.
Pakistan claims Mr Jadhav was working as a spy for India's external intelligence agency Research and Analysis Wing when he was arrested from the troubled province of Balochistan in March 2016. Days later, it had also released a video in which he was seen admitting to the claims. India has dismissed the video and refuted the allegations.