Several council members demanded the prize be abolished, while others called for it to be depoliticised. Fifa's football social responsibility committee will make a recommendation before the next council meeting in Doha on 16 December.
Infantino announced at the start of Thursday's meeting that the peace prize had been handed to the social responsibility committee. Council members returned to the subject later during a series of questions about possible conflicts of interest at Fifa.
There is a strong view among council members that the prize in its current form is tainted by its association with Trump. Some raised concerns that the US president could object and create a diplomatic storm if he disliked the identity of the next recipient.
The 22-person social responsibility committee is dominated by representatives from countries loyal to Infantino, so outright abolition may be avoided on the grounds that it would represent a humiliation for the Fifa president, although there is widespread acceptance that change is required.
One council member told The Guardian the most likely outcome discussed was the reinvention of the prize as an award for an individual within football who has delivered exceptional work promoting social harmony, with a provision stating it cannot be given to a politician.
Article two of Fifa's statutes prohibits political or governmental interference in football, one of several reasons why honouring Trump last December triggered a backlash from many members. Fifa announced the creation of the peace prize last November, saying it would reward individuals who had taken exceptional and extraordinary actions for peace and united people across the world. Trump received the award at the men's World Cup draw in Washington last December, with Infantino presenting a golden trophy, a medal and a certificate. The selection process has never been made public.
The social responsibility committee is expected to set strict parameters for any surviving award and to provide details of a nomination and selection procedure.