Following the release of court documents, Kuwaiti Sheikh Ahmad was linked in media reports to Richard Lai, the Fifa audit and compliance committee member who was banned for bribery.
Lai admitted taking $950,000 (£735,000) in bribes in a US Court on Thursday.
In a statement, Ahmad said he "vigorously" denies any wrongdoing.
The 53-year-old, a member of both Fifa's ruling council and president of the Olympic Council of Asia, had admitted on Saturday to being aware of the media speculation but said he was "very surprised" by the allegations.
Court documents in Lai's case did not not directly name Sheikh Ahmad but referred to someone who "at various times" was a "high-ranking official of Fifa, the Kuwait Football Association (KFA), and the Olympic Council of Asia (OCA)".
And in a second statement announcing his resignation, Ahmad said it was in Fifa's "best interests" for him to go.
"With regards to alleged illegal payments to Richard Lai, I can only refer to my previous statement and vigorously deny any wrongdoing," it read.
"I intend to work with all relevant authorities to disprove these, for me, totally surprising allegations.
"However, I do not want these allegations to create divisions or distract attention from the upcoming AFC (Asian Football Confederation) and Fifa congresses.
"Therefore, after careful consideration, I have decided it is in the best interests of Fifa and the AFC for me to withdraw my candidacy for the Fifa council and resign from my current football positions.
"I have been honoured to serve on the Fifa council, Fifa reform committee and AFC for the last two years and I will continue to support the family of football once these allegations have been disproved."
Lai, president of the Guam Football Association and a US citizen, admitted two counts of wire fraud conspiracy - relating to accepting and paying bribes - and one count of failing to disclose foreign bank accounts.
He agreed to pay more than $1.1m (£850,000) in forfeiture and penalties.
Following Ahmad's resignation, Fifa president Gianni Infantino stated: "I have taken note of the decision of Sheikh Ahmad Al-Fahad Al-Sabah.
"I want to thank him for taking this decision which certainly was not easy to take but is in the best interest for Fifa."