Ex-national security adviser Michael Flynn and his son discussed the alleged plot with Turkish representatives, NBC News and Wall Street Journal report.
The matter is said to be under scrutiny in the wider Justice Department probe into alleged Russian election meddling.
Mr Flynn quit after misleading the White House about meeting an envoy.
The alleged plot to remove the cleric, Fethullah Gulen, was first revealed in March 2017 by former CIA director James Woolsey.
The Turkish government accuses Mr Gulen, who lives in the US state of Pennsylvania, of being behind last year's failed coup.
He is viewed as chief political rival to Turkish President Recep Tayyip Erdogan, who has repeatedly called for Mr Gulen's extradition from the US.
According to the Wall Street Journal, special counsel Robert Mueller's investigation is focusing on a meeting in mid-December between Mr Flynn and Turkish officials in New York.
Mr Flynn reportedly discussed having Mr Gulen transported on a private jet to the Turkish prison island of Imrali.
Mr Flynn was serving on the White House transition team during the reported meeting, which came a month before he joined the Trump administration.
He also met Turkish representatives in September last year, according to Mr Woolsey, a board member for Mr Flynn's consultancy.
Mr Woolsey has previously told CNN that in September, "there was at least some strong suggestion by one or more of the Americans present at the meeting that we would be able, the United States would be able, through them, to be able to get hold of Gulen".
NBC reported that federal investigators are also looking into whether Mr Flynn tried to push for the return of Mr Gulen to Turkey during his time as White House national security adviser.
Mr Flynn's attorney did not immediately respond to a request for comment from the BBC.
But a spokesman for his company has previously denied he discussed any illegal actions with the Turks.
Barry Coburn, a lawyer for Mr Flynn's son, told the BBC he had no comment on the report.
Mr Flynn Snr was the first aide in Mr Trump's White House to resign, after only 23 days on the job.
The retired lieutenant general had admitted lying to Vice President Mike Pence about a meeting with the Russian ambassador in which the lifting of US sanctions was discussed.
Mr Flynn also failed to register as a lobbyist for the Turkish government while he was seeking White House security clearance.
In 2016, his consultancy Flynn Intel Group was paid $530,000 for lobbying on behalf of the Turkish government - work which required him to register as "a foreign agent".
His lawyer later said Mr Flynn did not register because he was working for a Turkish businessman, rather than a government official.
Investigators are also looking into the actions of his son, Michel Flynn Jr, who worked closely with him at Flynn Intel Group.
Both US media outlets also reported that Mr Flynn and the meeting participants discussed a way to free Turkish-Iranian gold trader Reza Zarrab, who is in a US jail over charges that he evaded US sanctions on Iran.