Speaking to the BBC, Sir Keir said he would “of course act” on Sir Laurie’s report which he expected to be “comprehensive” and delivered quickly but would not be drawn on whether Rayner would be fired.
He rejected a comparison with Boris Johnson, who as prime minister did not sack Priti Patel as his home secretary despite the then-adviser, Sir Alex Allen, saying she had broken the ministerial code.
The prime minister said he had strengthened the ministerial code and the role of the adviser since taking office.
The code sets out the standards ministers are expected to uphold including honesty and integrity.
Sir Laurie can advise on whether ministers have adhered to the code, but the prime minister decides what, if any, action to take.
Asked about the report, which sources say could be published as early as Friday, Sir Keir said: “I do think in the end we need to establish the facts, which the independent advisor will do and come to a conclusion.
“I don’t think it’ll take long now for that bit of process to conclude and then, of course, it does fall to me.
“I completely accept that, to make a decision based on what I see in that report.”
The prime minister has the power to dismiss Rayner from her roles as housing secretary and deputy prime minister but cannot remove her as deputy Labour Party leader because she was elected to the role by Labour members.
Sir Keir said he knew on Monday that Rayner was taking further advice on her tax payments.
