FAI chief John Delaney may be questioned about Fifa's €113m Irish company as well as the €5m pay-off after the Thierry Henry handball incident, a member of the Dail's sports committee has said.
The Dail's sports committee is meeting tomorrow, with members set to decide if they will invite Mr Delaney to appear before them.
The Herald last week revealed how Fifa established a company in Dublin that had Sepp Blatter as a director for seven years.
The firm, Fifa Ireland Ltd, received approximately €113m in revenue from the sale of licensing rights in relation to the 2002 World Cup and transferred around €112m of this to Fifa HQ in Zurich.
A number of FAI officials served as directors of the company but neither they, nor the FAI gained financially from their involvement, Fifa said.
Fifa said that its Irish business, which was dissolved in 2009, was "in full compliance with all relevant laws and regulations".
Mr Blatter (79) resigned as Fifa boss last week amid an investigation into corruption at Fifa. He denies any wrongdoing.
Mr Delaney was already expected to have to face questions from TDs on the €5m pay-off the FAI received from Fifa in the wake of the Henry handball incident.
Now committee member Patrick O'Donovan has said the issues in relating to the Fifa Ireland company should be probed as well.
"The committee is meeting tomorrow and we intend inviting the FAI in the very near future," the Fine Gael TD told the Herald.
"What we need to know is what links, if any, there were between Fifa Ireland and the FAI, what was the purpose of the company, what work it did, and how it was set up.
"If there is criticism of Fifa internationally we need to see how this company squares with that," said Mr O'Donovan.
Sport Minister Paschal Donohoe (inset) was asked about the possibility of Mr Delaney being quizzed by sports committee TDs.
"Whatever decision they make, I will be supportive of it," he said.
"It is important to make clear that this is not taxpayer's money that is being spent here," he said.
"We have very good mechanisms in place regarding the use of public money, the taxpayer's money, and all that is in place," he added.