Ben Emmerson QC is said to disagree with its chairperson Alexis Jay about its future.
The investigation is looking into the extent to which the Church and public bodies have protected children from sexual abuse in England and Wales.
Professor Jay was appointed to the position last month after previously leading an inquiry that revealed at least 1,400 children were subjected to sexual exploitation in Rotherham.
She is the fourth chair of the inquiry.
Mr Emmerson's departure would represent another setback for the probe, which has been beset by problems since it was launched by then Home Secretary Theresa May in 2014.
A spokesperson for the inquiry said Mr Emmerson has not resigned.
He is thought to want to restructure the group to help reduce the workload, according to The Times.
However Professor Jay said she has "no intention" of asking Home Secretary Amber Rudd to "revise or reduce our terms of reference".
The investigation was originally meant to last five years but there have been suggestions it could run for up to double.
Following her resignation, former chairperson Dame Lowell Goddard said there was an "inherent problem" in the inquiry's "sheer scale and size".
The inquiry spent £14.7 million out of a £17.9 million budget in 2015-16.
