JOIN USIndo-US relations and its future prospects figured in the discussions between Advani and Burleigh, sources said.
The situation in Pakistan also came up during the discussions.
The meeting was part of "routine consultations", a US Embassy spokesperson said.
Ahead of his meeting with Advani, Burleigh met senior BJP leader M Venkaiah Naidu a few days back.
The Indo-US civil nuclear deal and developments in Pakistan came up for discussion during that meeting, Naidu said.
"I talked about developments in Pakistan, growing threat of Taliban and gave my views," Naidu said.
"We are more concerned about Pakistan misusing American help and he said all our combined efforts should be to see that Taliban is totally vanquished," he said.
India also wants the US to put pressure on Pakistan to ensure that Pakistani soil is not used for terrorism against this country, Naidu said.
The BJP leader said he also conveyed his party's stand on the Indo-US nuclear deal.
"The US envoy keeps meeting prominent Indian leaders to discuss our views and exchange notes on issues like the nuclear deal, internal security and terrorism," Naidu said about today's meeting.
He said the US Charge d'Affairs informed Advani about the steps being taken by the US to control terrorism and discussed the growing clout of the Taliban in Pakistan.
"Till the US was not attacked they did not understand terrorism. They should understand the way Pakistan is misusing the funds, the help being given to Pakistan by the US, the money," Naidu said.
Burleigh had also met TDP leader N Chandrababu Naidu in Hyderabad a few days back.
The spokesperson said the meeting with Naidu was also part of "routine consultations" and had nothing to do with politics of the country.
"The US categorically denies any attempts to interfere in India's democratic political process," the spokesperson said while pointing out that some sections of media were projecting the meeting in a "distorted manner".