Mr Mousavi and Mr Karroubi were both presidential candidates in last June's disputed elections.
Anger at the result saw mass protests in Tehran and other cities that led to thousands of arrests and some deaths.
They have called for Monday to be a national day of mourning.
The grand ayatollah was one of Shia Islam's most respected figures.
He was an architect of the 1979 revolution that created Iran's Islamic state but later became a vocal critic of the government.
BBC religious affairs correspondent Christopher Landau says that in a country where the younger generation is often seen as increasingly secularised, Montazeri offered an understanding of Islam in sympathy with many of the opposition's concerns.
Christopher Landau, BBC News religious affairs correspondent
Grand Ayatollah Montazeri's ideas about religious freedom were particularly challenging to the current Iranian government.
Conversion away from Islam is illegal.
But Montazeri said that if a Muslim converted to another religion after careful study and using their free will, that choice should be permissible and respected.
In a country where the younger generation is often seen as increasingly secularised, Montazeri offered an understanding of Islam in sympathy with many of the opposition's concerns.
His death deprives Iran of an outspoken champion of an alternative vision for the Islamic Revolution, our correspondent says.
The cleric's death also comes during the sacred first month of the Islamic calendar. The 10th day of that month, the Shia festival of Ashura, is this coming Sunday.
It was already likely to offer a platform for opposition protests, and this death gives that date added significance, our correspondent adds.
On 20 December, following news of the grand ayatollah's death, a large crowd quickly gathered outside his home in Qom to pay respects.
Sporadic gatherings were also reported in Tehran and other cities.
Foreign media have been restricted in Iran since the post-election protests and it is often difficult to verify reports. No foreign media have been allowed to attend the funeral.
Supreme Leader Ayatollah Ali Khamenei did offer his condolences on Sunday, praising Grand Ayatollah Montazeri as an outstanding jurist.
MONTAZERI BBC INTERVIEWS
Exchanged e-mails detailing views on reform in Oct 2009
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