According to the Mail on Sunday, Lord Falconer made the comments during an online event hosted by a legal firm last June.
The Labour peer said he had been referring to legal questions arising from frequent changes to the law.
"But I very much regret my choice of words," he added.
According to the newspaper, he made the remarks during an online seminar hosted by law firm Gibson Dunn about law changes during the pandemic.
Lord Falconer, who served as justice secretary under former Labour prime minister Tony Blair, is a partner at the company.
In his introduction to the seminar, he is quoted as saying: "This is a gift that keeps on giving, the law keeps on changing, keeps on getting more complicated, and is always interesting."
'Poor choice of words'
The Labour peer told the BBC his comments "were expressly directed to the pace at which the government is making changes to the law and the legal questions that gives rise to".
Asked about Lord [Charlie] Falconer's remarks, Labour's shadow business secretary Ed Miliband told the BBC's Andrew Marr Show: "He shouldn't have said it.
"He was talking in the context of lawyers and the way the law was changing. But it was a very poor choice of words."
Mr Miliband said he had spoken to his colleague before appearing on the BBC programme and he was "very, very sorry and apologises for what he said".
"It shows that we have to be careful with our words, all politicians have to be very careful with our words," Mr Miliband went on.
"Charlie thinks the country has been through a terrible, terrible trauma and this in no way reflects his view about the Covid crisis."
As well as being justice secretary, Lord Falconer was a minister at the transport department, and then the Home Office, during the Blair era.
He went on to be appointed shadow justice secretary under the acting leadership of Harriet Harman, a role he kept under Jeremy Corbyn.
But he was among a dozen shadow cabinet members to later resign in the wake of the 2016 Brexit referendum over Mr Corbyn's leadership.
He was appointed shadow attorney general after Sir Keir Starmer's election as Labour leader in April last year.