Diana's sister Lady Sarah McCorquodale told the court she made the offer to Mr Burrell but he declined "because his memories were in his heart".
Lady Sarah said she had met Mr Burrell many times after the princess died, in August 1997, as they sorted through Diana's apartment at Kensington Palace.
Mr Burrell denies stealing 310 items from the princess, Prince Charles and Prince William.
He claims some were given to him by Diana because she disliked them.
Lady Sarah told the court: "I offered him anything he would like to take and he said he didn't because his memories were in his heart and that's all he needed."
Asked what sort of belongings she would expect Mr Burrell to have, she thought for a moment and replied: "Cufflinks, photographs in frames, enamel boxes, tie pins, ties - and I think that will be all."
Earlier, Diana's mother, Frances Shand Kydd, told the jury she had not known Mr Burrell had any of the late princess's possessions and that she had not given him permission to take any.
Giving evidence for the first time at the trial, Mrs Shand Kydd also said the former butler had misinterpreted his role in Diana's life.
The princess called him "my rock" but Mrs Shand Kydd said: "It is a term which she used for many people.
"She called me her rock and staff."
However, she also admitted she and her daughter had been estranged for the last four months of Diana's life.
Details of the reason for the rift were not disclosed in court but Lord Carlile, QC defending Mr Burrell, suggested it "arose as a result of an argument that you and she had had about her private life and the company she was keeping".
Presents
Asked by prosecutor William Boyce QC what Diana's attitude to security was, Mrs Shand Kydd said: "She was very, very careful with all things Royal, never did they pass through her hands."
He asked if Mr Burrell ever informed Mrs Shand Kydd he was retaining some property or sought permission to take possession of any items.
Mrs Shand Kydd replied: "never".
She was asked, in particular, whether he had ever asked to take a bronze ballerina, and replied firmly: "No".
Mr Boyce asked whether Diana had ever given gifts to her staff.
Mrs Shand Kydd replied: "Yes, she was very generous", adding that she never recycled gifts.
"I can promise you she gave away nothing other than gifts."
The court was told that Mrs Shand Kydd had shredded "50 to 100" documents including thank you letters to Diana after her death.
Lord Carlile asked whether she could remember if the destroyed documents had included thank you letters from Mr Burrell's wife Maria for clothes Diana had given her.
She replied that they did not, they were mostly from ladies-in-waiting.
The trial continues.