Fifa's ethics committee is looking into the "disloyal payment", made nine years after Platini carried out consultation work for Fifa president Sepp Blatter.
Platini, a Fifa vice-president, and Blatter are serving 90-day bans while corruption claims are investigated.
In September, criminal proceedings against Blatter began in Switzerland.
BBC Radio 5 live sports news correspondent Richard Conway says a written contract did exist for Platini's work for Fifa between 1998 and 2002, according to sources. A total of SFr 1.05m (£710,000) is believed to have been paid to the Frenchman.
It is an additional SFr 2m (£1.35m) that was paid in February 2011 that is under scrutiny - and for which no written contract is said to exist.
Both Platini and Blatter are believed to have told investigators the additional payment was an oral agreement for SFr 0.5m to be paid for each of the four years (January 1999 to June 2002) the Uefa president was employed by Fifa. In a statement this month, Platini said "the remuneration was agreed at the time".
It is understood the men claimed an agreement to delay payment was reached at the time due to Fifa's then financial position.
Under Swiss law, an employee cannot seek to have a contract paid up after five years have elapsed irrespective of whether it was written or oral.
Both men have denied any wrongdoing, have appealed against their bans and believe they will be exonerated.
Uefa, which meets in Nyon on Thursday to discuss the issue, has said it has "full confidence" in its president and "stands fully behind him".
However, one Uefa executive committee member, Allan Hansen, has been quoted by Danish newspaper Ekstra Bladet as saying "we can no longer support him" - if no contract exists.
Meanwhile, Fifa has banned Thai football chief and former executive member Worawi Makudi for 90 days for an ethics breach and indicated he could face further punishment.

Angelika and Marcin Klis had not been seen since the explosion at the Ariana Grande concert on Monday night.
The couple's daughter Alex Klis, 20, a student at York College, had posted an appeal on Facebook for information about their whereabouts.
The Polish foreign ministry confirmed two Polish nationals had been killed.
It is understood Mr Marcin, 42, worked at York Cars Taxi Service, a cab company in York.
Mr Marcin and his wife, aged 39, had taken a picture of themselves together in Manchester shortly before going to the concert to collect their daughters.
This was shared on social media by their daughter on Tuesday.
She wrote: "Anyone who is in any safe place or hospital in Manchester, if anyone comes across my parents please please let me know as they've been missing ever since the attack, this is a picture taken tonight so this is exactly what they were wearing."
In a statement, the Polish foreign ministry said: "We share the pain of the family and the loved ones of the victims and express our most sincere condolences.
"Consuls have offered assistance to the family of the victims."
The Polish Foreign Minister, Witold Waszczykowski, told a radio station: "The parents came after the concert to collect their daughters and unfortunately we have information that they are dead. The children are safe."
Dr Alison Birkinshaw, principal at York College, described the deaths as "devastating news for Alex and her family, the whole college and the wider community".
She said: "We are in touch with Alex and will do whatever we can to support her and her family."
A JustGiving page has been set up to raise money for Ms Klis and her sister Patrycia and has so far raised more than Â£4,000.
Uefa chief Michel Platini produced no documentary evidence for the 2m Swiss francs (£1.35m) payment made to him by Fifa, the BBC has learned.
[NEXT_CONCEPT]
A Polish couple living in York who had gone to Manchester Arena to pick up their daughters have been killed in the terror attack.