Shaikh Salman, the most powerful man in Asian football, says "senior" figures in the game have told him to run for the Fifa presidency.
In a letter to members of the Asian Football Confederation (AFC) executive committee, the 49-year-old Bahraini said he is not yet a candidate.
But he said he had been urged to stand "by a growing number of senior football administrators, Fifa members and personalities of public life".
The election takes place in February.
The deadline for candidates to submit the relevant paperwork - together with letters of support from five football associations - is 26 October.
Sepp Blatter will step down as president of world governing body Fifa in February.
Shaikh Salman, the president of the Asian Football Confederation, publicly backed Uefa boss Michel Platini for the Fifa presidency in July.
But he now appears open to stepping forward himself following the Frenchman's 90-day suspension.
Human rights group Americans for Democracy and Human Rights in Bahrain has written to Fifa asking it to prevent Shaikh Salman from standing, alleging "unethical behaviour" while he was president of the Bahrain FA.
|Profile: Shaikh Salman Bin Ibrahim Al-Khalifa|
|Aged 49. Born in Bahrain. Cousin of the King of Bahrain.|
|President of the Asian Football Confederation since 2013. Former head of Bahrain FA.|
|Chaired Fifa committee that decided dates for 2022 Qatar World Cup.|
|Supports Manchester United and is a keen hunter of wild boar.|
|Faced human rights questions over arrest of Bahrain footballers involved in pro-democracy protests during the Arab Spring.|
|Denied Bahrain FA had been involved in "non-footballing activities".|
Platini, 60, is suspended as Fifa investigates reports that he received a £1.35m payment in 2011 from Blatter, who is also suspended. Both deny any wrongdoing.
In his letter of 18 October, Shaikh Salman said he had not "actively considered" running and added: "More importantly, I am not a candidate today."
He also urged AFC executives to consult with him and give their views, insisting he will not make up his mind until he has heard from them.
"You and the AFC membership that you represent are the people that I am responsible to and whom I am elected to guide and serve," he wrote.
"Without seeking your views, I would fail in my duties. It is largely in your hands if I accept the challenge."
Shaikh Salman said he would also take advice from close friends, advisers and his family before deciding whether to be a candidate.
His AFC presidential campaign was overshadowed by allegations that he was complicit in human rights abuses during pro-democracy protests in Bahrain in 2011, something he strenuously denies.
Meanwhile, an "extraordinary" meeting of Fifa executives is expected to confirm on Tuesday the presidential election date of 26 February will remain unchanged.
The ongoing reform process is also expected to be discussed, with Francois Carrard - the chairman of the Fifa committee examining various options - due to present an update.