He was put on leave on Thursday after allegations he was implicated in a scheme to sell World Cup tickets.
Valcke, 54, who denies any wrongdoing, has three years left on his contract at football's world governing body.
Meanwhile, Fifa president Sepp Blatter has told colleagues he will not leave Switzerland during investigations.
Fifa has been engulfed by claims of widespread corruption since May, when Swiss police raided a hotel in Zurich and arrested seven of its top executives.
United States officials have since indicted those seven and two other Fifa officials on bribery and racketeering charges.
Newspaper allegations on Thursday implicated Frenchman Valcke in a scheme to sell World Cup tickets for above face value.
He now faces a formal investigation by Fifa's ethics committee.
The BBC has contacted Valcke's lawyer and is awaiting a response.
It is understood Valcke, who has held the position since 2007, wanted to be paid out in full for the remainder of his deal. A senior Fifa official described Valcke's demands as "ludicrous".
His predecessor, Urs Linsi, was reportedly given a pay-off worth £3.6m.
Prince Ali bin al-Hussein of Jordan, who is standing in next February's Fifa's presidential election, said in a statement: "The reputational damage to Fifa continues. Surely it is time for a new generation with new ideas to take over.
"The old regime should step aside now, so a new team can effect a complete reform of the organisation to reflect the change that the national associations and fans of the game around the world are calling for."
Valcke, who last month considered standing for the Fifa presidency, is Blatter's number two at the organisation.
Blatter, who announced that he would stand down just days after winning re-election in June, has emailed Fifa staff saying the organisation can recover from this "difficult situation" and "restore its reputation for the good of the game".
The 79-year-old Swiss stated in July he would not take "travel risks" until his legal position is clarified.
In that same month he attended the 2018 World Cup qualifying draw in St Petersburg.
Blatter was due to be in Moscow on Friday to attend a celebration marking 1,000 days until the start of the Russian World Cup, but opted to stay in Zurich.
Fifa sources have told the BBC that he has been advised by his US lawyers to stay in Switzerland.
The decision not to go to Moscow is believed to be unrelated to any fear of arrest. Russia has no bilateral extradition treaty with the US.
Blatter is due to hold a meeting of Fifa's executive committee in Japan at the end of the year but there are now questions over whether that will proceed as scheduled.
Earlier on Thursday Eugenio Figueredo, one of the seven officials arrested in May, had his extradition to the United States approved.
As well as the US inquiry, a Swiss investigation is looking into the bidding process for the 2018 and 2022 World Cups.
Fifa has recently set up a taskforce to tackle corruption which aims to "restore the integrity and reputation" of the organization.