The review, headed by its director of editorial complaints Peter Johnston, will look at the allegations against Brand and what was known about them during his time at the BBC.
The comic worked for BBC radio between 2006 and 2008 as well as appearing on other shows.
In a question and answer session with staff, Davie said they were dealing with “very serious allegations”.
Brand has strongly denied the allegations, which also include claims of controlling, abusive and predatory behaviour.
Four women made sexual abuse allegations against the star between 2006 and 2013 as part of an investigation by The Times, The Sunday Times and Channel 4’s Dispatches published at the weekend. Police have since said they have received an allegation of sexual assault said to have taken place in Soho in 2003.
Mr Davie said: “I think it’s absolutely right that we’ve said, along with organisations, that we urgently look at the issues that have been raised, and that goes across a number of organisations including the BBC.”
He told staff there would be “full transparency” with the review “digging into anything that we’ve got there”.
The review will also also examine Brand’s use of a BBC car with one of his alleged victims telling journalists he sent a BBC car to take her from school to his house when she was 16 and he was 30.
Mr Davie said: “Peter will report to the Executive Committee and the Board, and we want to complete this work as swiftly as possible. I’m hoping for an initial report in weeks, not months. So this is not something we want to take too long, but we do want to get to the facts.
“And just to emphasise, because people do ask, the objective is to be totally transparent, just share what we what we have, and be really supportive in terms of how we do it.”
The review is said to look “at any complaints against Brand, what was known, and what was done”.
The BBC earlier announced it was removing some content featuring Brand from its iPlayer and Sounds apps which "now falls below public expectations".
An episode of QI and a Joe Wicks podcast, both featuring Brand as a guest, have been removed, it is understood.
While the boss of Channel 4 says the broadcaster has "carried out extensive document searches and thus far have found no evidence to suggest the alleged incidents" about Russell Brand were reported to management.
The BBC is currently in the middle of reviewing how it dealt with the scandal around presenter Huw Edwards.