Thompson, who is on holiday, came to the decision after the furore over the offensiveness of the pair's calls to Andrew Sachs grew with Prime Minister Gordon Brown saying he wanted the BBC to take "appropriate action".
After a campaign to sack the pair led by the Daily Mail, the BBC has received more than 18,000 complaints, with around 8,000 coming overnight.
Pressure grew this morning with the girl at the centre of the pair's comments, Sachs' granddaughter Georgina Baillie calling for them to be fired.
Thompson said the pair had been guilty of a "gross lapse of taste" and added his own personal apology.
He has ordered shows featuring Brand and Ross off-air until the case is resolved.
The first show affected is an episode of 'Never Mind The Buzzcocks' featuring Brand, scheduled for broadcast tomorrow night on BBC Two.
Ross' chat show 'Friday Night with Jonathan Ross', which has six more shows in its run left, will not air this week.
Also affected is BBC Radio 2's Saturday schedule with both Ross's morning and Brand's evening shows suspended.
Thompson's statement read: "I would like to add my own personal and unreserved apology to Andrew Sachs, his family and to licence-fee payers for the completely unacceptable broadcast on BBC Radio 2.
"BBC audiences accept that, in comedy, performers attempt to push the line of taste. However, this is not a marginal case. It is clear from the views expressed by the public that this broadcast has caused severe offence and I share that view.
"Since Sunday, I have been in regular contact with the senior executives I tasked with handling this issue. The investigation that I instructed Tim Davie to conduct is nearing completion, and I am returning to London to review the findings and, in the coming days, announce what action we will take.
"In the meantime, I have decided that it is not appropriate for either Russell Brand or Jonathan Ross to continue broadcasting on the BBC until I have seen the full report of the actions of all concerned.
"This gross lapse of taste by the performers and the production team has angered licence payers. I am determined that we satisfy them that any lessons will be learnt and appropriate action taken. I have been asked to report to the Trust's editorial standards committee before the end of this week and will discuss with the Trust the findings of the report and the actions I propose."
A spokesman for the BBC Trust said that it had asked the director-general to report the findings of management's investigation and the actions he proposes to a special meeting of the Trust's editorial standards committee tomorrow on Thursday where Sir Michael Lyons will be present.
"The Trust expects to return to this issue again at its next editorial standards committee meeting on 6th November and its full Trust meeting on the 20th. However, the chairman will make a statement about the Trust's initial reactions before the end of this week."