Mr Dunn died when his motorbike crashed into a car outside an RAF base on August 27 of this year.
The suspect, Anne Sacoolas, 42, was granted diplomatic immunity following the crash.
The Foreign Office has said in a letter that it no longer considers that Mrs Sacoolas has diplomatic immunity.
However, the Home Secretary refused to be drawn into whether she thought Mrs Sacoolas could now be extradited.
When asked about extradition on BBC1’s The Andrew Marr Show Ms Patel said: “The Foreign Secretary has been working with his American counterpart, he has been in touch with the US administration on this.
“It very much seems that the lady in question wants to start co-operating with the discussions and the investigations and I think that we should support that.
“We need to ensure that justice is done but obviously that co-operation with this investigation takes place. That is absolutely right.”
This follows the release of a letter written to Harry Dunn's family by Foreign Secretary Dominic Raab.
The BBC, who have seen the letter, reported that it said: "The US have now informed us that they too consider that immunity is no longer pertinent."
It reportedly added: "We have pressed strongly for a waiver of immunity, so that justice can be done... Whilst the US government has steadfastly declined to give that waiver, that is not the end of the matter.
"We have looked at this very carefully... the UK government's position is that immunity, and therefore any question of waiver, is no longer relevant in Mrs Sacoolas' case, because she has returned home."