The governing body wrote to the outgoing Rangers chief executive more than two weeks ago and has received a response, but it is still looking for more clarification.
Green stepped down last week following claims by Whyte, although he denies any wrongdoing.
Whyte, who was banned from football for life by the SFA after his failure to pay tax saw Rangers go into administration, claims he was behind the company which was initially set up to purchase the assets of the liquidation-bound club in June last year.
Green denies working as Whyte's front man and claims he only told Whyte what he wanted to hear in order to secure a deal to buy the former Rangers owner's shares, which were subsequently not needed as a new company was set up.
Rangers have employed law firm Pinsent Masons and financial investigators from Deloitte to probe the alleged links.
This week's developments included a Rangers statement to the stock market admitting that Sevco 5088, the company at the centre of the dispute, was their subsidiary when they were floated on the stock exchange in December. The statement also confirmed that Green was a director of the company.
A different company, Sevco Scotland, ultimately secured the assets following the £5.5million purchase before the name was changed to The Rangers Football Club.
Whyte has claimed he was a director of Sevco 5088 and a company working on his behalf this week sent information over the dispute to the Serious Fraud Office.
Meanwhile, Green's SFA hearing over his alleged use of "racist and offensive language" in a newspaper interview has been scheduled to take place on May 2. Although Green has resigned as chief executive, he remains under SFA jurisdiction as a director of the club.
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