Schofield has already apologised for the incident, which occurred last week on the This Morning programme, but he was later disciplined by ITV. Ofcom has also launched an investigation.
None of the names on the list, compiled by Schofield after "trawling the internet", were read out on the programme, but ITV has apologised after a "misjudged camera angle" meant that some of them may have been briefly visible to viewers.
Last week, Lord McAlpine secured a £185,000 settlement from the BBC after the corporation's Newsnight programme ran a report that led to him wrongly being linked to alleged child abuse at a North Wales care home.
He could have got much more from the BBC, but his spokesman said that he was "very conscious that the money was coming from the taxpayer".
But this is not the case with commercial broadcaster ITV. A source close to the former Tory Treasurer told The Daily Telegraph: "With ITV they are planning to get considerably more than the BBC because they are a commercial organisation."
Lord McAlpine's lawyer Andrew Reid is now busy targeting everyone who linked his client's name with the alleged abuse, including threats of lawsuits against thousands who mentioned his name on Twitter.
A number of high profile people, including George Monbiot and Sally Bercow, are allegedly in the firing line.
But Reid has also singled out This Morning's Schofield for particular criticism, after he warned David Cameron that there could have been a "paedophile ring among the elite of Great Britain that led all the way to Downing Street".
Live on air, Schofield then handed over a list of names of senior Tories who were the subject of child abuse allegations on the internet.
"Phillip Schofield managed to embarrass the prime minister as a side part and then destroy my client's reputation," Reid told BBC Radio 4 last week.
"What he did was very, very low and I am amazed it was allowed. It sent everyone on to the internet - those who couldn't read what was there - to see who being referred to. At the top of the list was Lord McAlpine."
Schofield later apologised for the incident, and insisted that he was not taking part in "any kind of witch hunt".
"Unfortunately there may have been a misjudged camera angle for a split second as I showed the prime minister some information I had obtained from the internet," he said.
"I asked for his reaction to give him the opportunity to make a point which he very clearly made about the dangers of any witch hunt."
ITV's director of television Peter Fincham stated that he had spoken to Schofield, who is said to be "under no illusions that this was a lapse in ITV journalism; this is something we shouldn't have done".